This is a really interesting question and I can't wait to see an answer. As a secondary question - was the entire empire similar to Rome in terms of diversity, or would a region like Hispania or Britannia be predominantly populated by locals with a Roman presence?

I can't give a full answer but I will put on general guidelines.

First we have to distinguish various areas, some are way more diverse than others. The most importants are the big metropolis like Rome or Alexandria. For exemple we have traces of tombs of people who have died in Rome but who came from various regions of the Empire, like the Middle East with Syrians or Jewish people. Another exemple is the attested textual presence of Indians in Alexandria, I never heard of Indian objects found there but in Berenike on the red sea, the last port on the road to India we have found fabrics that were made with an indian technique, proof of ship repairs made by Indian sailors (it's a very recent discovery made by new excavations of the city), some probably going to Alexandria another part may travel by caravan across the desert to the Nile. On top of this Alexandria was know for it's important jewish community, I know nothing about nubian inhabitants but their total absence seems quite improbable. Last but not least the Ptolemies imported many galatian celts there, we have trace of celtic languages using the greek alphabet during the ptolemaic era but I don't know if it was still around much later. That celtic language was still speaked in Galatia (central Anatolia where most galatian settled in the late 4th early 3rd century BC) in the 4th century AD as noted in a letter of Sint Jerome.

In the levant who already had influcts of various groups the greek was the main language but there were still semitic languages in use like Syriac. On top of that roman administration added Latin (even if Greek was used as the common language) and more importantly roman colonist as exemplified in the plain of the Bekaa in Lebanon. Now don't forget that behind roman colonists there could be romans citizens at the end of their service. And roman auxiliaries recieved roman citizenship and piece of land to work, so it certainly helped to spread some groups across the empire. Does someone know more about a possible relation between the region of origin of the legionnaries and the localisation of the land they received?

The spread of various group of popultion was certainly not homogenous, a bit in the same way there are more or less diverse area in the US. People were attracted by big cities of by the land they were offered. You would have no reason to move to the countryside of a remote area to buy land in a society in which information moved much more slower.

What call besides trade did most foreigners have to travel to Rome? Was it viewed as a place where a person could find opportunity to establish a life? Was there much upward mobility?

Just for reference, how does the population of the Roman empire at it's height compare to that of new York? Would the methods used be as reliable as those today, i.e. is there a discrepancy in the record?

I would assume that slaves (who presumably were mostly war prisoners?) considerably changed the population of Rome and the surrounding latifundia?