2005 Dodge Grand Caravan: is it my battery or alternator?

I have an 05 Dodge Grand caravan that I got used from my grandma. I play in a band and in the past couple years I’ve taken the van on 2+ cross country 5000+ mile tours. It’s been pretty reliable.

Recently, someone broke in and tried to hotwire the van. I found it with the steering column cowling ripped off and the lock cylinder battered with a brick, but they were unsuccessful thanks to the anti theft system. Also, when I found it, the battery was dead. After some research, I convinced myself I needed to replace the immobilizer anti theft unit, because the would-be thieves had smashed the ring antenna of my old immobilizer. I managed to splice the circuit board from the old immobilizer into a new one I bought, and I used my roommates Prius to charge my battery and start the car.

This was about two days ago - I successfully drove the van to work on that day (a ~45 minute drive). However, when I got off work about 4 hours later, the battery was dead again. I work on a college campus and a campus cop was nice enough to help me jump the battery so I could get home.

However, on the way home, I pulled over to make a phone call and put the car in park for a minute. After some time, the lights on the dash started to flicker and I started to feel the engine sputter and something made a kind of grating noise. I gave it some gas and started driving again, and the problem went away.

Fast forward to today, I try to start the van to go to work and the battery is dead again.

My question is - do you think this is a battery problem, on account of the car is ageing (170,000 miles), or an alternator problem? Or, is it a combination of these? Or it it neither, and I just need to drive it for a long time to charge the battery back up to full?

Totally discharging a battery is bad for the battery. You could have it charged then tested to see how much life is left in it. If it is more than 3-4 years old it probably needs replacing.
As far as the alternator, many automotive stores like Autozone will test it for free.
The other thing to do is, with the battery charged have the parasitic draw checked, ie how many amps are being used when the vehicle is off, after the work you did there might be something drawing the battery down.