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

As others have said, you won't need a military passport to study abroad, but you should get clearance from your command. There are also different policies and procedures for different branches and depending on where you want to study, and there are some special security concerns for military personnel overseas as we can often be targeted in certain locations. It's also important that your unit has full accountability for you at all times. One of my LTs just did a semester abroad in Paris and was living near the concert hall where one of the recent terrorist attacks occurred. Because he was briefed and prepared, we were able to maintain accountability and offer him assistance if he needed it.

It is all dependent on the Reserve participation scheduled designed by your Command and the blessing from your Chain of Command (CoC). If you discuss it with your leadership, adding the benefit to both you and the Unit once completed, you will likely be allowed to do some stacked drill before or after your 3 month sojourn. I would be hard-pressed to see the downside to this unless there is a mandatory exercise or mission support Annual Tour during this timeframe.

Irrespective of that, congrats on the amazing opportunity! I hope it all works out since something like this is more complicated to into life as one gets older.

I don't think this should be an issue unless it is going to cause you to miss your annual training. If you are not I would just talk to your CoC and inform them of what you are wanting to do. It is likely they'll approve it and you will just have to make up the drill dates in advance or after you return (depending on how your trip falls within the fiscal year). Also, unless you are going on a trip for the military on official orders you will need a civilian passport, not a military one.