Most who claim Warfield as a theistic evolutionist owe their position to the analysis of Mark Noll and David Livingstone, who together published a selection of Warfield's writings on the subject in a compendium called Evolution, Science, and Scripture. Noll wrote his analysis in a portion of his book Jesus Christ and the Life of the Mind, reprinted at BioLogos, and Livingstone did so within his book Darwin's Forgotten Defenders. So it would serve us well to examine the writings of Warfield that they themselves cite. Fred Zaspel, while critical of the "Livingstone-Noll thesis," nonetheless says in a Themelios article that the compendium "collects virtually all of [Warfield's] creation-evolution writings."
If you want to examine the evidence yourself, the best primary sources are those in Evolution, Science, and Scripture, and the best secondary sources would probably be Noll's and Livingstone's writings in favor of their thesis and Zaspel's against.
It's important to note that Noll and Livingstone claim development within Warfield's thought. The following appears both in the Noll link above and in their joint introduction to the compendium:
In the course of his career, both Warfield’s positions and his vocabulary did shift on the question of evolution. But they shifted only within a fairly narrow range. What remained constant was his adherence to a broad Calvinistic conception of the natural world — of a world that, even in its most physical aspects, reflected the wisdom and glory of God—and his commitment to the goal of harmonizing a sophisticated conservative theology and the most securely verified conclusions of modern science.
The nuance is best picked up on by reading their editorial comments throughout the compendium, but for our purposes the development is not extremely important, except to note the following. There is no documentation of Warfield's views before his first published work on the subject in 1888, other than his 1916 personal recollections on undergraduate life at Princeton, speaking of one of his teachers:
He did not make me a Darwinian, as it was his pride to believe he ordinarily made his pupils. But that was doubtless because I was already a Darwinian of the purest water before I came into his hands [in 1868], and knew my Origin of Species and Animals and Plants under Domestication, almost from A to Izard. In later years I fell away from this, his orthodoxy. He was a little nettled about it and used to inform me with some vigor—I am speaking of a time thirty years agone!—that all biologists under thirty years of age were Darwinians. I was never quite sure that he understood what I was driving at when I replied that I was the last man in the world to wonder at that, since I was about that old myself [early 1880s] before I outgrew it.
So it would seem that he accepted Darwin's theory for much of his young life, "outgrowing" his pure acceptance of it sometime around the early 1880s, but never fully rejecting its possibility in his published works post-1888. Phrased in this careful way, I don't think even Zaspel would disagree.
In an effort to present the strongest case for Warfield as a theistic evolutionist, I'll arrange this answer topically, using your bullet points as a guide, with two additions of my own:
- Acceptance of the usefulness of the term "evolution"
- Antiquity of the universe and/or of the human race
- Adam not a historic, individual person
- Adam a historic individual, but a product of evolution from a lesser mammal
- Death of mammals prior to the Fall
- The Fall not a discrete historical event
We'll examine one at a time whether he affirmed these ideas.
Warfield on "evolution"
Warfield's 1895 article "The Present-Day Conception of Evolution," a two-part essay which is largely a reworking of his 1888 lecture "Evolution or Development," says the following at the opening of part two:
There are three general positions which may be taken up with reference to the doctrine of evolution, which has so deeply affected modern thought as to the origin of the universe and all that it contains.
We may look upon this doctrine as supplying an obviously true and adequate philosophy of being, and treat it as furnishing a complete account of the origin and present state of the universe. ...
We may consider the doctrine of evolution as a discovery by science of the process through which this ordered world in which we live has, as a matter of fact, come into existence; we may treat evolution merely as an account of the manner in which the universe, considered as a cosmos, has been produced, and all the forms of being which constitute it have been brought into being. In this form, evolution ... is made a second cause and implies a first cause working by and through it. ...
We may look upon the doctrine of evolution as a more or less probable, or more or less improbable, conjecture of scientific workers as to the method of creation; and thus we may treat it as only a working hypothesis suggested to account for the manner in which the universe has come into being, and seeking now to try itself by the facts. This has always been the attitude of the more cautious thinkers.
Warfield counts himself one of the "cautious thinkers" who consider evolution a "conjecture" and "working hypothesis." After describing the three possible positions, he first dismisses the first position, as he had already spent part one doing, as "the old problem of the atheistic philosophy" in "a new dress." Then he breezily says that, were evolution proven true, "the old faith will be able" to "assimilate to itself all facts," concluding thus:
The only living question with regard to evolution is whether it is true. And the only reasonable reply which can be given to this question today is that it is sub judice [before the court].
He similarly concludes his 1888 lecture by stating that one can be Biblically faithful and accept evolution:
I do not think that there is any general statement in the Bible or any part of the account of creation, either as given in Genesis 1 and 2 or elsewhere alluded to, that need be opposed to evolution. ... There is no necessary antagonism of Christianity to evolution, provided that we do not hold to too extreme a form of evolution. To adopt any form that does not permit God freely to work apart from law and which does not allow miraculous intervention (in the giving of the soul, in creating Eve, etc.) will entail a great reconstruction of Christian doctrine, and a very great lowering of the detailed authority of the Bible. But if we condition the theory by allowing the constant oversight of God in the whole process, and his occasional supernatural interference for the production of new beginnings by an actual output of creative force, producing something new, i.e., something not included even in posse [potentially] in the preceding conditions, we may hold to the modified theory of evolution and be Christians in the ordinary orthodox sense.
Note his caveats, that even if one accepts evolution, one must accept God's miraculous intervention in order to still accept the Genesis account. In his 1911 article "On the Antiquity and the Unity of the Human Race," he goes further and implies that, not only does a purely evolutionary (non-miraculous) account of human origins defy the Bible, it also defies common sense:
Men seemed to imagine that, if only time enough were given for it, effects, for which no adequate cause could be assigned, might be supposed to come gradually of themselves. Aimless movement was supposed, if time enough were allowed for it, to produce an ordered world. It might as well be supposed that if a box full of printers' types were stirred up long enough with a stick, they could be counted on to arrange themselves in the order in which they stand, say, in Kant's "Critique of Pure Reason." They will never do so, though they be stirred to eternity.
These threads come together in the conclusion to his 1901 essay "Creation, Evolution, and Mediate Creation":
What, then, is to be the attitude of the Christian man toward the modern doctrine of evolution? He is certainly to deny with all the energy given to him that the conception of evolution can take the place of creation as an account of the origin of the universe. Evolution offers no solution of the question of origins. For its operation it presupposes not only already existent material which can unroll into fresh forms, but material within which all that is subsequently evolved already potentially exists.
And he is to deny with equal strenuousness that the conception of evolution can take the place of that of mediate creation, as an account of the origination of new things in the course of the divine government of the world. Things have come into being since the first origin of the world which did not lie potentially within the primal world-stuff, needing only to be deduced from it. If nothing else, the God-Man has come into being; and that not as a product of the precedent conditions in the world, but as an intrusion from without and above. And with him, the whole series of events that constitute the supernatural order of the Kingdom of God. Nor is there any reason to doubt that the same intrusion of purely creative force, productive of something absolutely new, may have occurred also in the natural order of the first creation — say at the origin of self-conscious, immortal beings in the complex of nature.
On the other hand, the Christian man has as such no quarrel with evolution when confined to its own sphere as a suggested account of the method of the Divine Providence. What he needs to insist on is merely that Providence cannot do the work of creation, and is not to be permitted to intrude itself into the sphere of creation, much less to crowd creation out of the recognition of man, merely because it puts itself forward under the new name of evolution.
Age of the universe and humanity
Warfield, in his 1911 article "On the Antiquity and the Unity of the Human Race," says that the Ussher chronology and similar attempts to discern the age of creation or humanity are misguided. He argues at length that the genealogies are not meant to be any sort of clue about how much time elapsed between Adam and Abraham. He concludes that the Bible does not tell us the age of mankind, and says thus:
The question of the antiquity of man is accordingly a purely scientific one, in which the theologian as such has no concern. As an interested spectator, however, he looks on as the various schools of scientific speculation debate the question among themselves.
He goes on to state that there is considerable debate between the sciences of geology, biology, and physics over the age of the creation, while granting that for the age of the earth the question regards how many millions of years.
The same general argument is traced out in his 1903 article "The Manner and Time of Man's Origin."
Individuality of Adam
Warfield argues in "On the Antiquity and the Unity of the Human Race" that, contrary to the ideas of racists, Scripture and science are united in claiming a singular origin point for humanity rather than multiple origins for different races. In my reading, he heavily implies Adam as the single origin.
In his review of James Orr's book God's Image in Man, Warfield explains his view this way:
We partake in Adam's sin because he was our representative. ... He was constituted our representative because he was our father and as such was naturally indicated for that office.
Animal ancestry for humanity
Warfield seems to accept the theoretical possibility of pre-Adam "brutish" proto-humans without too much fuss. In his review of Orr's book, he invokes what we now call "punctuated equilibrium" to reject Orr's argument that the body and mind must have evolved together if evolution is true:
Body and mind must go together [argues Orr], and a great brain with a little mind is just as unthinkable as a little brain with a great mind. The argument does not seem to be available, however, as against a theory of evolution per saltum [by leaps]. If under the directing hand of God a human body is formed at a leap by propagation from brutish parents, it would be quite consonant with the fitness of things that it should be provided by his creative energy with a truly human soul.
Similarly, in his article "Creation, Evolution, and Mediate Creation," he admits the possibility that "Adam's body was formed (not created) from the lower animals," while insisting that each human being's soul, Adam's included, is a special creation of God.
Death before the Fall
Warfield speaks about pre-Fall death strangely little, but perhaps my wish for him to have addressed it more is anachronistic. The only mentions in the index of Evolution, Science, and Scripture of "death" occur in his review of Orr's book. All he says regarding the issue in question is this:
Perhaps also a query may be placed over against the strong statement to the effect that "there is not a word in Scripture to suggest that animals ... came under the law of death for man's sin." The problem of the reign of death in that creation which was cursed for man's sake and which is to be with man delivered from the bondage of corruption, presses on some with a somewhat greater weight than seems here to be recognized.
It's ironic that the only statement Warfield makes on this subject is a complaint that another author doesn't deal with it adequately.
The nature of the Fall
In both "Evolution or Development" and in his review of Orr's book, Warfield decries a common (even today) evolutionary depiction of the Fall as representing awareness of sin. He argues in both places that Scripture depicts the Fall as a fall from morality into immorality, and that the idea of the Fall as a development of a "moral sense" within humanity or of human beings from "nonmoral beings" into "moral beings" reverses the Scriptural picture. As far as I can tell, he never directly addresses allegorization of the Eden account, instead always treating it straightforwardly.
Warfield believed that the theory of evolution was neither proven nor disproven in his time and that it was permissible for Christians to believe a limited form of it. Like modern theistic evolutionists, he believed that the age of the universe was an open question and that it was possible that Adam had evolved from "lower animals," though he affirmed Adam's historicity and the special creation of his soul. His thoughts on the Fall and on death before the Fall are less clear, but he doesn't seem to give credence to views that make the Fall less than historical, and he's cautious about accepting pre-Fall animal death.