4age hitting gas too quick means engine stalls - what is the issue?

[Bean]

Well the light issue is entirely unrelated as mentioned before.
At those temps chances are right that the starter is stuck, happens fairly quickly (also in sandy environment not just cold), this is assuming your battery is still good of course 12V+ (13 would be even better)

You should here a solenoid click from the passenger side when trying to start. Even when the starter doesn't turn at all or actually even more so as when it works you can barely hear it over the sound of the starter.

PS: you could try push starting the car, if it starts driven around for 30min this should warm up the starter enough to unfreeze but not too much to overheat.