2005 Toyota Corolla intermittently won't start then stalls after 1 mile

Sounds like an electrical issue. The no start can be due to a weak battery, or a poor battery contact. Sometimes there will still be enough juice to power the lights but not enough to crank the starter.

Having it die while driving could be from the same kind of issue, but also points to a a charging system, since that should be giving more than enough power while the engine is on, even if the battery is not that great. And if the charging is poor, that could actually be causing the issues with the battery in the first place.

Weather can be a factor too depending on where you are. Metals contract in cold, and sometimes this can be just enough to make the battery contact (or some other contact) too poor to conduct properly.

At minimum, I would check the battery terminal contacts to see if they are loose, and clean the contact surfaces if you can. Also check the battery voltage and alternator voltage. If you don't have a meter yourself, any reputable shop should be able to do this. (probably at no charge, it should only take a couple of minutes).

You mention the alarm which shouldn't normally be a problem, but if the battery is already flat it's possible that just having the alarm could push it over the edge. It's also possible, though rare, that some parasitic drain from the alarm is what is causing the battery issue in the first place.