What was Byzantium's relations to Roman myths (such as Romulus, the Aeneid, etc.) like?

The Byzantines loved all things Roman, and mythology - if it was sometimes harder to get at, whether because of its morally dubious themes or simply a language barrier - was no exception. The Byzantine taste for mythology is nearly ubiquitous in art and literature, but it can be hard (at least for moderns) to distinguish what of this is Greek or specifically Roman. Fortunately, though, there's a relatively long tradition of Virgilian reception in Byzantium that might help answer the specific case of the Aeneid.

It's hard to measure premodern "popular culture," but Virgil was one of the most important Roman poets, even in the Byzantine period. Part of this is due to the Fourth Eclogue's supposed prediction of Christ - Constantine quoted most of this poem in a (Latin) Easter speech, which Eusebius of Caesarea then translated into Greek - but it's not the only reason he's so popular. The Aeneid was one of the most widely-used educational texts in the late empire. Papyri from Egypt show that students both read the Aeneid in the original Latin as well as made their own rudimentary Greek translations. Even when Greek started to overtake Latin in Constantinople, familiarity with both Greek and Latin remained the defining characteristic of an educated person. Eusebius isn't the only non-Latin author to quote Virgil: John Lydus (writing in Greek) cites him at least five times, and he's the only Augustan poet mentioned by John Malalas (writing in Syriac). The Baths of Zeuxippus, located immediately beside the Great Palace of Constantinople (and very close to the Hagia Sophia) had a number of statues depicting historical and mythological figures, including Aeneas, Creusa, Dares, and Antellus - and even Virgil himself. A Greek poem describing these statues addresses Aeneas directly ("shield-bearing offshoot of the land of Troy," whose "piety announces wondrous descent from golden Aphrodite") but also does the same to Virgil ("clear-voiced swan, dear to Italians...whose native Echo of the Tiber raised as another, Roman Homer"). Nor did Virgil's popularity decrease after Latin had indisputably died out: Virgil pops up a surprising number of times in the Suda, a 10th-century encyclopedia (as does Aeneas in, for example, this absolutely bizarre entry). As time goes on, Byzantine knowledge of Virgil does tend to get more and more fantastic - much like medieval western ideas of Virgil as some kind of immortal, all-wise magician - but they never forget that he was the Roman poet, and that, as Romans, they have a corresponding affinity for him.

I'm most familiar with the case of Virgil, but he isn't the only Latin, pagan poet this happens to over the long course of Byzantine history. Maximos Planoudes (ca. 1260-1305) translated Ovid's Heroides and Metamorphoses (among other things) into Greek, but he's a bit late, after the Latin conquest of Constantinople anyway, and is not known to have ever translated Virgil. While the Byzantines certainly enjoyed, say, Polybius or Plutarch (Greeks who wrote about Roman things as essentially strangers), recurrent examples like this indicate the Byzantines really liked the idea of Roman mythology, even after it stopped occupying the place in their culture it had in pagan antiquity.