Is study abroad possible for reservists currently in school, and are there possible hurdles besides missing drill?

Sgt (Join to see) - Like the others stated, get clearance from your command. BUT, I do not recommend that you get a government passport (unless, of course, you're going on official business). Get a regular civilian passport. No need to "flag" yourself for being anything other than American.

Per the MCRAMM you must submit a formal letter via your command to MFR. I'm not currently reading the reference but I believe it states that you can request to be drop to the IRR if you're overseas stay is longer than 30 days. Look it up or ask your 1stSgt.

Unless it is military related, it will not involve a military passport. Write a letter to your commander and security manager expressing your intent to study abroad. I taught English in Japan for a year when I was a captain, and arranged to work with a unit at Camp Zama, though I was still assigned TPU status in California. I would thing that three months should not be a big issue, as long as your command is tracking. You can RST before and after the three month tour, but remember, the RST must be written in the interest to support your unit. I am sure your command will be happy to have you in for a few days to work on weapons inventory or equipment maintenance prior to departing for the three month study abroad.