Whether you can marry depends on your state law. Your immigration and his being on probation shouldn't get in the way of you getting married.
However, your immigration status is another matter entirely.
As a permanent resident, he could petition for you (his wife), but you would be subject to the yearly cap - and have to wait years for your priority date to come current before adjusting status (being eligible for your own green card). You check your priority date by looking at the Visa Bulletin. As a spouse of an LPR you'd be category 2A, which currently looks like at least a 4-5 year wait. I'm guessing your education would be done by then.
Ordinarily I would suggest your boyfriend consider becoming a citizen. However, he is on probation, and depending on the crime(s) he's committed, I would definitely hire an immigration lawyer to evaluate if his immigration status is in trouble.
If he cannot apply for citizenship, you many want to look into getting permanent residence through another avenue, like employment.
I wish you both the very best of luck.
Response to this question is for informational purposes only and does not constitute a client/attorney relationship. A client/attorney relationship is created by signing the retainer agreement. For a more complete answer to your question or individualized advice, please hire an immigration attorney.