I'll edit and answer the first part when I'm back home with my books (I'm away from home on a business trip). Until then, I'll just answer the second bit. But note on the first part that there was also a large jewish community in Mesopotamia.

How many of them were converts (or were most descendants of Judeans/Israelites)?

It's difficult to say for certain. We know from Josephus that conversion took place in Roman areas, and the Talmud implies it. But there's no clear ratio of converts to native Jews.

Also, did they practice rabbinical Judaism or did they perform sacrifices and have a priesthood? If the latter, was it connected to the priesthood/temple in Jerusalem or were there other temples?

There's a bit of a false dichotomy here. Rabbinic Judaism does have sacrifices and a priesthood. The former is on hold while there's no temple to perform them, but the historical belief of rabbinic Judaism was (consistently until the 19th century) that sacrifices are not void. The liturgy is based off the sacrificial system, and references it, including references to its future resumption. The priesthood still exists, just with a much-diminished role without sacrifice.

Anyway, answering the denominational makeup isn't really possible either. Unless you were a religious authority involved in a particular group (namely Pharisees or Sadducees were the two big ones, but there were more. Assuming a rabbinic vs non-rabbinic religious division in late ancient Judaism is often assumed, but isn't quite how things were), you were probably pretty ambivalent. Josephus said that most of the Judean common people were pro-Pharisee, but rabbinic texts generally state that the peasantry were not really "rabbinic", even if the formal structure and religious authorities were pharasaic. But that goes mostly for Judea itself. Elsewhere, outside the centers of the religious institutions, it's pretty likely that people would be similarly apathetic towards the religious devides, even if they formally practiced something resembling one sort.

There is record of another temple at elephantine in Egypt, but well before the time period you're asking about. By this time there wouldn't have been secondary temples .