“Seeing as there is only one mediator between God and man, what’s the role of the prophets?”
The mediatorship of Christ, which is absolutely unique, is not exclusive to all other mediatorship, in the context of the passage, only exclusive of other Reconcilers with God in the ultimate sense.
1 Timothy 2:1-6 (DRB) I desire therefore, first of all, that supplications, prayers, intercessions, and thanksgivings be made for all men: 2 For kings, and for all that are in high station: that we may lead a quiet and a peaceable life in all piety and chastity. 3 For this is good and acceptable in the sight of God our Saviour, 4 Who will have all men to be saved, and to come to the knowledge of the truth. 5 For there is one God, and one mediator of God and men, the man Christ Jesus: 6 Who gave himself a redemption for all, a testimony in due times.
Rather than His Mediatorship (albeit unique by nature, and exclusive in this sense alone) precluding any mediatorship of others (e.g. Job 42:8; Rev. 5:8; 8:4; etc.), this passage uses the mediatorship of Christ as the basis upon which intercession of others, who are not Christ, is made meritorious or effective. Contrary to any notion that all intercession or mediation is rejected here, it teaches that participiation in (not usurpation of) Christ's mediatorship is the will of God: "For this [intercessory prayer] is good and acceptable in the sight of God our Saviour, who will have all men to be saved, and to come to the knowledge of the truth." It would be a contradiction in terms, were we to say Christ's mediatorship excludes all other mediatorship just after saying we should intercede on behalf of others—that misinterpretation must give place to the prior affirmation of other forms of mediatorship in verse 1.
What is meant, then, is that Christ, being the incarnate Word, is the ultimate Source of reconciliation for mankind—"the mediator between God and man." The Mediator of the Covenant of Redemption (Heb. 9:15). What it cannot mean, due to the many instances of approved mediation (including one glaring example, the priesthood) is that his mediatorship precludes others' mediation in other ways.
It is in this sense that exclusivity is denoted by "one" (as in "one" mediator between God and man).
Another way of looking at it might be that priests themselves (one might say, the highest form of mediatorship outside that of Christ's) themselves are in need of reconciliation with God, whereas Christ is the Reconciler of all in Himself, being God and man Himself, uniting God and humanity in the ultimate sense in His own person. Priests in a sense mediate salvation, but so do prayers for conversion in a way also by laypersons, but clearly this is quite different from God reconciling man to Himself in the person of Jesus Christ, true God and true man.
“To me it seems clear that all the promises of God are in legal force for all who are in Christ, but it’s not because they were spoken by the prophets, but rather because they were spoken by God. The prophets were merely listening in and relating what they heard.”
This is quite true in a sense, but authorities carry authority in their own right, something Jesus teaches quite clearly. They have authority from God, to be sure, but nonetheless they possess this real authority with which they have some freedom to exercise it as they will.
Matthew 23:1-4 (DRB) Then Jesus spoke to the multitudes and to his disciples, 2 Saying: The scribes and the Pharisees have sitten on the chair of Moses. 3 All things therefore whatsoever they shall say to you, observe and do: but according to their works do ye not; for they say, and do not. 4 For they bind heavy and insupportable burdens, and lay them on men's shoulders; but with a finger of their own they will not move them.
I don't know if this authority comes quite reaches the level of "whatever you bind on earth will be bound in heaven" (Mt. 16:18; 18:18), or "whose sins you forgive they are forgiven them; whose sins you shall retain, they are retained" (Jn. 20:21-23) the authority given to the Apostles, but it is quite able to be used at the discretion of those who wield it—even, as can be seen in Jesus' example, to bind what are unfair or heavy laws (lowercase 'L') on subjects to said laws, and even if they are, hypocritically, not obeying it themselves.
“Yet God says that He does nothing without the prophets. That sounds more like a choice by God than a legal requirement that we can’t talk to God without the prophets.”
The fundamental reason there are prophets and messengers is simply that "no man can see God and live" (Ex. 33:20; cf. Jn. 1:18). Even God appearing in a visible form (Isa. 6:1; cf. Jn. 12:41) would be an "angel" or "envoy" (מלאך) of His own presence (Isa. 63:9), you might say.
“Being “in the image of” seems to be related to the biblical idea of human authority on the earth, but it seems to indicate far more than just decision making authority. It goes toward the idea that we should have a resemblance of God and people that know Him should say that we remind them of Him.”
Indeed, if God is willing to save humans it must certainly be that the image of God which is in them by nature represents something of His own nature, and as such is of a very high, quasi-infinite worth. How can else can we explain God finding need to give mercy to people who all hate Him? There must be something wort restoring that He saw in them, something much greater and more profound than simply 'given the decision making authority on earth.' "In the image of God He created them" (Gn. 1:27) is essentially saying 'God put something of Himself in them.' I think this is where man's great authority comes from and cannot really be separated from meaningfully.