I'm currently in Boston, as an international student, and can definitely appreciate the housing stresses! I've also lived in Canada, where we'd usually see places going up 30 days before, but in Boston I'm asked months in advance (usually 3-4) if I'll be resigning my lease. There is a short period of answer turnaround, and then the place is listed if I don't resign. This does vary a bit place to place, but this seems to be reasonably standard for real estate companies, whereas private landlords seem to post places closer to the vacancy date (this information is solely based on my own personal experiences and those I've seen from my friends and colleagues).
I had a lot of luck renting with Copley. They didn't ask me for much documentation, just ID and my social security card, although a friend of mine was able to sign a lease when she explained she didn't have a SSN. They did require first and last months rent plus a brokers fee, but no damage deposit. All of those payments had to be made using a certified check from my bank, so you'd need to have those funds in your bank account to acquire the check.
If you're looking to get the lay of the land before you dive into the renters market in Boston (and gather al your funds together!), I would suggest subletting for a few months. This is what I did, and it made the transition a lot easier. Just be aware that move in day at the end of August is a city-wide event, and it can be a difficult and chaotic time to acquire a lease. I'm part of a bunch of MIT/Harvard subletting groups, feel free to PM me if you'd like me to send you the links!