Mix of things there.
Firstly on paying for his ski pass or whatever it is... no no no. Cheeky fucker!
Paying half of petrol - yes, if you otherwise paid equally for the kids and you both wanted them to do this. Then it would be fair. I don't think a divorced couple can divvy up every expense to the penny, but major expenses should be reviewed individually and in principle I think it's fair to consider splitting petrol costs. But - you have considered it, and the swim trips balance it so - nothing owing.
The time... well, if he only has the Sunday (?) and they're going to miss 10 weeks of access, then I'd be open to swapping to do 5 of the days. Not because of the time or the cost, but because it's a long time not to see their dad properly. And in that case, you'd get the petrol cost too. If you really can't afford that, then it's fair to say you'll swap 5 of his Sundays but you need the petrol money. He seems OK about missing the time though.
The car... when you split, you have to let go of shared ownership. Have you completed your financial settlement? If he's using the car and you're not, you need him to buy you out of it. It just isn't sustainable to think of it as also yours. In an amicable split where you can say 'could I borrow your car to do x?' then fab - but this is not amicable.
Amusing to think of him stressing that you might leave crisp crumbs in his precious... but rude if you do it. Even if it really is jointly owned, if you have decided to let it stay with him, it's rude to not give it back clean.
Can't believe he has the cheek to ask you to pay half for him though!!!! Just a simple "no" is the answer- no reasons and sadly no letting rip 
just "no".