If you mutually agree with your current employer to vary the terms of your contract, I would say it's a question of whatever you agree. If they want rid of you sooner than the end of 13 weeks then they can offer to pay you off.
Again, you're entitled to whatever your contract says re the company car - unless you negotiate something different.
I'd have a good read of your contract.
If your current employer is acting reasonably, I wouldn't push too hard. Better to leave on good terms.