Jaiye, send him a certified letter giving him so many days to pick up the dog as per the court's order or he forfeits any and all rights to it. The worst he could do is file contempt but the judge would probably rule against him since he's had two months already and still hasn't picked up the dog. You could also file contempt as well, claiming any expenses related to having and caring for the dog since the day it was awarded to him. But I'd just wait and let him file it, then file a counter to it and ask for reimbursement.
MamaKitty, your ex has no right inside the home uninvited since he no longer lives there. I assume that you've been awarded temporary possession of the house? He also has no right messing with the car, either. Shoot, I'd rather pick up a carseat off the porch rather than hassle with unhooking it or hooking it back up when I'm done with it. I think it has more to do with invading your privacy than it does the carseat.
As for where kids are exchanged, I've done it at several places - school, fast food restaurants, my parents' house, her parents' house, her house, my house, her friend's house, inside, outside, in the middle of the road even (ok, we pulled into a parking lot, but you get the point lol). If you're not comfortable with him being at your house, agree to take the kids to him and pick them up, or agree to meet somewhere else. If he says no, then just request that the court order the exchanges take place elsewhere and explain that he's coming into your home uninvited and hanging around for no reason whatsoever.
Finally, the dog situation. Most courts consider pets as property even though many people treat them like they were kids (furkids, if you will). Since you got the dogs after marriage they would be considered marital property. But if you can show you've taken the most care of them and are still taking care of them, it's doubtful he'd be given one. Even the kids appear to have more sense than him in telling him to just get another dog, so obviously he's just using that as another way to aggravate you. Teach the dog to bite him and he might leave it alone lol...
Anyway, just my opinions...