OP I feel for you.
You sound really nice, and like you want to be kind and not hurt MILs feelings or cause any hurt feelings or confrontation.
Much like you I'd feel sick at the idea of asserting myself with MIL about the driving and the 'sleepovers'.
However, it's definitely best nipped in the bud with clear and direct truthful statements. These can be diplomatically put so as to try not to hurt feelings, but please don't get yourself tangled in a web of excuses as she'll fend ways to meet anything you can come up with.
Re. sleepovers it is perfectly acceptable to say you will not be ready to leave DD for a long time (you don't have to specify whether it's 'anyone' or 'MIL' you won't be happy leaving her with).
TBH I think kitting out the nursery at hers is a really pushy way of steamrollering you into doing something you're not ready for. It might help to think of this if you have to assert yourself to her face to face.
Re. driving. Wait and see what husband comes up with, but it's also completely ok to say something like 'you've had a few accidents and near misses recently and I don't feel comfortable letting you drive DD, especially whilst she's so tiny, but I want her to bond with her granny so let's find some other ways to make this happen'
Don't apologise for any of it! You are only don't what's best for DD and that's what parenthood is all about.
And once you've said your bit you can stop stressing about it.