The Divine Comedy wasn't meant to provide a literal description of Hell (or Purgatory, or Heaven). The literary milieu Dante drew on to write the Comedy was incredibly complex, so it's only natural that the end product is equally sophisticated. While certain elements, like notable Florentines or juicy legends, were doubtlessly included to entertain and edify his audience, a great deal of the poem - the system of punishments/rewards, the classical references, even the metrical/philological structure of the poem itself - can instead be read allegorically to reveal a deeper, "mystical" meaning.

• Punishment/reward: Punishment in the Inferno is never random; it reflects the sin that earned it. Epicureans (atheists, or those who denied the immortality of the soul) are stuck forever, as souls, in the burning sarcophagi of Dis (Sixth Circle). Those ruled by violent impulses spend eternity submerged in the Phlegethon, a river of boiling blood (Seventh Circle). Judas and Brutus (traitors against Christ and Julius Caesar) are condemned to be chewed up by Lucifer (a traitor against God) for the rest of time (Ninth Circle). This principle (contrapasso) dates back to Aristotle, but was also developed by Thomas Aquinas in a Christian context to explain how sins were punished. This is also true of Purgatory and Heaven to a certain extent. In Purgatory, for example, the lustful are punished by being made to stand in burning fire, but because the souls know it's ultimately good for them, they willingly submit to it; sinners in Hell offer their unsolicited opinion quite often, but in Purgatory, Dante has to go out of his way to talk to people. In Heaven, every soul is infused completely with joy, but the intensity of the joy varies according to the soul's proximity to God, which is determined by their primary virtue (zeal, love, contemplation, etc.). This synthesis of classical logic (proportion) and theology (punishment of sin/rewarding of goodness) thus fits nicely with the Comedy's vision of a harmonious, ordered Christian universe: the entire cosmos is based on justice.
• Classical references: On the one hand, this is simply part of Dante's education. Like almost any literate medieval, he was taught to read by being taught Latin; naturally this would have consisted substantially of wholesome scriptural and moral works (the Bible, scriptural commentaries, even a few approved pagan moralists like Seneca or Cicero), but even in the Middle Ages people still appreciated things like Virgil (seen as the Latin Homer) and Ovid (prolific mythographer, but kind of racy love poet). Dante himself reveals his great respect for the Latin poets by putting them among the Virtuous Pagans (to say nothing of Virgil's job guiding him through two thirds of his overall journey). It's also important to note, though, as you mentioned, that most of this pagan imagery is in fact reserved for Hell. Centaurs like Chiron and Nessus can guard sinners in Hell, but so do demons like the Malebranche. As soon as we get back into good, Christian territory all of this changes. Charon ferries the damned across the Acheron into Hell, but an angel ferries saved souls across the ocean to Mt. Purgatory; in Purgatory, the guards are all angels, not creatures of classical mythology. There's still discussion of antiquity in Heaven - when Dante gets to the Sphere of Mars, for example, he talks a lot about Aeneas and Scipio Africanus - but it's never as present or real as the Christian counterpart (e.g., Dante's crusader ancestor, Cacciaguida). Dante's love for classical antiquity is immense, but in the end, it's still superseded by his love for God - Dante openly criticizes numerous Popes (for which he certainly may have gotten in trouble) but he does so on account of what he sees as sin.
• Literary structure: Some of this is still evident even in translation. The entire Comedy consists of 100 cantos divided into three canticles: Inferno (34), Purgatory (33), and Heaven (33). Of these, only Hell is imperfect; the other two are numerologically significant (3*10 +3). In the original Italian, the entire poem (14233 lines!) is written in concatenated terza rima - ABA, BCB, CDC - which continues the trinitarian theme throughout. The quality of the language changes, too, as Dante progresses. Hell has moments of ribaldry and dialectal Italian that would be completely out of place anywhere else (the demon Malacoda "made a trumpet out of his ass" in Inf. 21.139); Purgatory has the only example of a non-Italian language (other than Latin) in the entire poem (Arnaut Daniel's speech at the end of Purg. 26); while Heaven is full of abstract, Latinate passages, often said to be the most beautiful of the entire poem. All of this reflects the increasing perfection of Creation as Dante gets closer to God - another major theme of the Comedy as a whole.

Of course, the Comedy is certainly meant to entertain. Dante intentionally imitates the classical poets and includes exciting mythological references because that was the high literature of his time. But it's also meant to educate. The universe is a comedy, according to Dante, because it rests entirely on God, the ultimate good. Centaurs guard in Hell while angels guard in Purgatory because Hell is for the wicked and Purgatory is for the good: neither group has any business switching places with the other. Dante's reworking of classical references in a non-pagan, non-Latin context is no accident - instead, it's part of his overall achievement.

Recommended further reading:

• The University of Texas' Danteworlds site is a great, readable introduction to the all the references and arguments hidden in the Comedy, organized by canticle/canto. Excellent because it's free, but also great for portability (most other works on Dante are pretty hefty).
• Charles Singleton is possibly the standard English commentator on Dante; the entire Comedy, with his translation and commentary, is available from Princeton University Press.
• Lino Pertile's "Introduction to Inferno" (Cambridge Companion to Dante, 2nd ed., 67-90; Cambridge, 2007) gives a great overview of how Dante uses both contrapasso and pagan mythology in the Inferno, specifically. A. N. Williams, in the same volume, has an article on Dante's theology ("The Theology of the Comedy," 201-217) if you're really interested in the nuts and bolts of the topic.

Not really. As /u/arivederlestelle points out, he was both showing off his education and being allegorical with the classical references. Also, he putting the pagans and classical monsters right where a Christian would expect to find them: Hell (except for Cato). He's not presenting them as beings to be worshipped by any means, which would be heretical.

It was also perfectly acceptable in Christendom to admire the good people who weren't Christians just as they never deluded themselves into thinking they were neccesarrily better because they were baptized. (One of Dante's points is that the Church of the time was focused on Italian politics to the exclusion of preaching the Gospel and calling sinners to repentance.) The conduct of the Muslim Saladin, for example, was used to contrast with the acts of Christian princes and a staple of the chansons de geste that were popular were the Nine Worthies -- three pagan, three Jewish and three Christian knights who exemplified chivalric ideals.

It's a bit off topic, but you could almost say that many of the differences between pre- and post-Reformation Christianity in the West are summarized by the way Dante and Milton treat Satan. Milton is full of sympathy for the Devil, depicting his pride as a kind of Greek hubris while he's not even a character in Inferno, clearly doesn't reign in Hell and is just as impotent as any other sinner.

Just to pile on to what /u/arivederlestelle has said in his fantastic answer - Dante's language and imagery are richly allegorical, and not necessarily to be taken as something the poet literally believed. One source of imagery that hasn't been mentioned here is astronomy, which seems to inform a lot of the structure and content of Dante's poem.

As for the issue of allegory, Dante perhaps explains it best in his own words:

"To elucidate, then, what we have to say, be it known that the sense of this work [(viz. the Divine Comedy)] is not simple, but on the contrary it may be called polysemous, that is to say, 'of more senses than one'; for it is one sense which we get through the letter, and another which we get through the thing the letter signifies; and the first is called literal, but the second allegorical or mystic. And this mode of treatment, for its better manifestation, may be considered in this verse:

'When lsrael came out of Egypt, and the house of Jacob from a people of strange speech, Judaea became his sanctification, Israel his power.‘

For if we inspect the letter alone the departure of the children of Israel from Egypt in the time of Moses is presented to us; if the allegory, our redemption wrought by Christ; if the moral sense, the conversion of the soul from the grief and misery of sin to the state of grace is presented to us; if the anagogical, the departure of the holy soul from the slavery of this corruption [(i.e. the bondage of the soul within the body)] to the liberty of eternal glory is presented to us.

And although these mystic senses have each their special denominations, they may all in general be called allegorical, since they differ from the literal and historical; for allegory is derived from alleon, in Greek, which means the same as the Latin alienum or diversum.”

-Dante Alighieri (Letter to Can Grande della Scala, 7)