Is this to a center with lots of other kids or just one or two others? In either case I don’t think I’d be taking a favorite toy, or any toys at all as I would think they already have them. You’ll risk losing them or just leaving them there when you’re in a hurry.
Again I’d leave it at home, they will either find something else to latch on to, or someone else may want to latch on to it. Kids love to take things away from one another.
I haven’t a clue as both our kids are in places where they wouldn’t ever need a car seat. You’ll have to ask about that one.
I think the biggest thing you’ll have to worry about is leaving them. More for you then them. They’ll cry and scream and it’ll break your heart, but you just have to leave them. They will get over it. Hell I was just told my oldest didn’t want to go back to school after two weeks off, though I’m sure that 10 minutes later she’s all good.