What is the current understanding of the specifics of classic Egyptian afterlife beliefs?
That is impossible to answer without going through an entire shelf of books. Egyptian ideas of the afterlives changed significantly over time, certain regions and social groups also had their own specifics
Who would and would not come to live again after death, for instance?
Even poor people, who could not afford mummification, statues, a full formal ritual, still received burials with at least modest grave goods and small brick niches for offerings. So I think that people hoped for an afterlife even if they didn't have the full force of magical equipment and rituals. Some wisdom texts (Merikare, Ptahhotep) stress the judgement before the gods as the decisive factor, others warn that a well equipped tomb is needed or weep for those whose bodies are lost and cannot receive a burial.
So I believe that the Egyptians themselves would have had some heated debates on the topic, rather than an uniform opinion.
Would they return to their previous bodies or receive new ones?
Both. The Ka and Ba (two of the types of soul) could return to the original body, if it was mummified (a "sAH"). Or they could return to take residence in statues and images of the deceased... but not to live there permanently, mostly to come visit and receive offerings.
Was it expected that they would resume living on Earth (specifically Egypt, I assume)?
No, in the Duat, or in the Beautiful West, the realm of the dead. But it remains ambigous as to where this place is located.
In the Old Kingdom, ideas of the afterlive were still more cosmic and talked about becoming stars and travelling with Ra in the sun barque. In the Middle and New Kingdom it shifted to more Chthonic imaginations of caverns and underground realms.
Any noteworthy interactions with other religions?
They influenced the levantine region in the late Bronze age, as it was pretty much their colony and they had some mines, fortresses and administrative centers there. Interactions with other Bronze age superpowers could be quite diplomatic, (when they weren't currently at war, that is)... for example, sending divine statue that was supposed to have healing properties, to another kingdom.
They also adopted some levantine deities into their pantheon, most notably Ana and Astarte.
A special relationship existed with the southern regions, the Ku****es and Nubians (these terms would need a more precise definition, but that would take some time). Anyway, Egypt colonized and exploited them for quite a long time, but in the Late Period got invaded by them in turn. A thriving Egyptian/Ku****e mixed culture and religion survived in Meroe even after Egypt had fallen to the Greeks.
Culturally, what is known about their view of themselves and their relationship with other communities, particularly the Rome and its colonies?
By the time the Romans became a relevant force, Egypt was already ruled by the Ptolemaic dynasty and integrated into the hellenistic world.
The Egyptians of the Bronze Age had a weird double standard when dealing with foreigners. In their spiritual system, anything outside of Egypt was foreign, chaotic, outside of Ma'at, and therefore needed to be kept out and destroyed. There's tons of ****ty racists pharaonic propaganda about subjugating the "wretched foreigners". But in reality, they were rather pragmatic. They did trade expeditions, had diplomatic relations, travelled, and had foreign colonies (well, not in the way the Romans did it. The Egyptians never tried to establish their culture and administration in foreign lands. It was more about maintaining military control and exploiting resources. )
