I don't recall reading anywhere if the keys were left in the car either. I also wonder if Tony might have asked her about taking the car seat out, because she would have had to use it later.
I can't imagine that he would have. Guys, that age and without kids, don't really think about that kind of thing. Also, Caylee hadn't been around for a while at that point (IIRC, the guy who was staying with Tony while he waited for his own apartment said he last saw Caylee there 6/13) - she was "with the nanny/with the grandparents" per Casey, so there wasn't a pressing need to have the seat.