The driver races on foot obviously

It's more of a "let's review what you spent last year" than a rolling totalization. So if a team was really that concerned they could abstain from a race, but I imagine they have stuff budgeted much better than that, even the shit show teams

No, they wouldn't be given an extra budget. That would defeat the purpose of the cost cap. If they go over it they will get penalized accordingly. Spare parts are part of the cost cap for a reason. If a team knows they would go over the budget by building a spare part they have to make the decision on whether to take the penalty or not participate in the race.

I tried to find out the exact dates but I couldn't find it so I have to make assumption. I just highly doubt the financial season of the cost cap ends with the last race. I assume the season will roll over some time later. So any team would budget a proper safety buffer for spare parts which they can just burn in development after the last race. And adjust their budget for mid-season development if there have been more crashes than anticipated. In a normal season spare parts from crashes make up less than 5 mil, which is about 3.5% of the cost cap.

Also don't forget that a lot of the more expensive parts teams still have used ones lying around. If an engine blows in the last 1-2 races I doubt they will build a new one and just re-use an old one. Same with gearboxes, power units, and some other parts. Especially if they are aware they are uncomfortably close to the cost cap.