What does it mean for something to be theologically sound?
Ordinarily, something is considered theologically sound in reference to something. Dante wrote in late 13th and early 14th Century Italy where like most Italians of his day, he would have been Catholic and surrounded with a strong Catholic culture. Thus, the question has to be how sound is the theology expressed in Dante relative to Catholicism, especially as would be understood in his day. To compare otherwise is nonsensical because Protestantism as such did not exist.
Dante's Comedy is harmonious with Catholic theology; it just isn't meant as a theological treatise and thus has many details which serve as literary details not as conveying literal theological truth.
Consequently, much of the implication that Dante's theology is not Biblical or unsound is reducible to the same arguments Protestants make against Catholic theology, which often are based upon misconceptions of Catholic theology.
Dante certainly wasn't a priest, but like many educated people in his day, he did study with priests; Dante studied philosophy with the Dominicans, an order of priests which places high emphasis on preaching and academic study. In Dante's day, they were newly established, but today 800 years later they teach and preach throughout North and South America, Europe, Africa and Asia.
The Divine Comedy has been called "the Summa in verse" referring to the slightly earlier work Summa Theologiae of St. Thomas Aquinas, which outlines systematically Catholic theology and explains the justifications for it, drawing upon a variety of sources including Scripture, the Church Fathers (especially Augustine), philosophers (especially Aristotle) and commentators (e.g. Peter Lombard). The Comedy expresses in a poetical and literary from much of the theology of the Summa in a narrative form, using characters, images and tropes from the Classical tradition.
The question of the eternal fate of virtuous pagans and babies is a complicated theological point. Not even all Protestant schools of thought agree (as an aside, I seem to recall that Calvin dismisses the possibility entirely of virtuous pagans being in heaven).
The Catholic Church teaches as dogma that "souls who depart from this life in the state of original sin are excluded from the Beatific Vision of God" which would ordinarily include unbaptized infants and the virtuous pagans (the non-virtuous pagans like their baptized counterparts would be punished in Hell). This doctrine follows in part from John 3:5 "Unless a man be born again of water and the (Holy Spirit) he cannot enter into the Kingdom of God" (as cited in Fundamentals of Catholic Dogma by Ludwig Ott, pg. 125)
Dante's depiction of Limbo matches with the theological theory prominent in his day. Modern theologians sometimes express other possibilities for the fate of unbaptized infants at least in part. Dante describes Limbo as a place at the top of Hell and lacking in hope, which approximates the poena damni (pain of loss) due to lack of the Beatific Vision. Leaving aside the case of infants, the problem with assuming "well-meaning pagans" would go to heaven on account of still being governed by the law is that it negates the purpose of the crucifixion and makes grace part of the natural order, an error known as Pelagianism, which was rejected by the Church in the 4th-6th centuries, especially by St. Augustine.
Dante's depiction of Suicide likewise follows from the Catholic tradition especially Aquinas. The association between Mental Illness and suicide is relatively recently understood to the point that that association was used to justify softening the Church's policy in the 20th century around ecclesiastical funerals for suicides. It's certainly not fair to critique Dante for something he nor anyone else really knew about until the past century. Additionally, just because some or even many suicides are related to mental illness doesn't mean that suicide categorically is morally permissible nor that no suicide bears responsibility for his actions. Only God can truly say whether a soul who commits suicide did so willfully or not and thus assign the soul to heaven or hell.
In Book I of the City of God by St. Augustine, Augustine comments on two well known suicides from Roman history: Cato's and Lucretia. While it's impossible to rule out any mental disturbance from underlying, the circumstances as preserved in the historical accounts (which yes may not be entirely factual) suggest in both cases willful intent. Cato basically committed suicide so that he wouldn't have to live being defeated by Caesar. Lucretia committed suicide out of shame from being raped by the son of Tarquinius Superbus (the last king -- this rape and suicide precipitates the war leading to the expulsion of the kings and the establishment of the Roman Republic). Dante clearly takes a softer view on Cato than Augustine because he places Cato not with the suicides but at the base of Purgatory. Ultimately, God alone knows the state of the soul.
Both of the answers relating to a Protestant viewpoint correctly point out the Comedy was not meant as a doctrinal treatise but as a literary work (it should be further noted that it is also not meant as an account of a mystical experience unlike many other works in the Catholic tradition). Yet, these answers conflate literary details (things which are meant to tell a story) with a theological claim. The Comedy like many works in the Medieval and Classical canon uses elaborate symbolism to tell the story and relate theological or philosophical concepts. The structure of Heaven, Hell and Purgatory are not meant as Dante's literal view of what those three realms are like but a mix of symbolism to reveal the theological concepts and literary flourish to make it compelling. For example, the shape of Hell roughly corresponds to the division of evil into Incontinence, Brutishness, and Malice deriving from Aristotle with the particular sins ordered within each division according to a progression for least bad to more severe. The structure of the poem and the realms described therein are also filled with numerical significance with 3, 7, and 9 being prominent.
None of that means that Dante meant to convey that Hell, Heaven or Purgatory are in fact shaped like or operate exactly as he described.
The Protestant critique cited in both answers completely misses the point of the narrative and its theme. It's not about "salvation by works" and that's not what Purgatory is for either. Nor is Purgatory about a second chance after death. As far too common, Protestant critique of Catholic works and theology amounts to a more superficial engagement with the concepts. The Comedy is about a souls progression from sin towards God through detachment from love of evil (sin), purifying the will towards Good and spiritual growth in holiness.