In answer to your main question...maybe not. I understand the frustration, but I'm sure it wasn't the ex's intention/desire that the car seat get soaked, either. Things happen. I'm sure if this were typical, you would have said so. I also assume you're more concerned about whether the kid's diaper gets changed as often as it should when she's away from home, than about having to wash the car seat pads this one time. Lashing out and telling the ex she needs to buy her own car seat isn't going to make her a more responsible parent, if in fact she does ignore the girl's diapers.
It's fine, if people want to have a different car seat for each car their kid rides in frequently. Personally, I've never understood it. Who's using the car seat in your car, while your kid's riding in the one in Grandma's car? Are car seats really so hard to remove? I think switching our big, clunky stroller around is more of a pain. If the ex prefers to use the car seats you've already bought, I don't think it's such a big deal. If it were snowy and you had snow boots for the kids, but Mom didn't, wouldn't you just send them with the girls when she picked them up, instead of making a point that Mom had to buy her own?
In answer to your overall question: "Do you share car seats with the NCP?" - That makes me laugh (now). My twins are almost 15. Car seat laws have gotten much stricter, since they rode in them. Once they grew out of the infant seats, my ex thought booster seats were ridiculous. I used to dutifully take the twins' boosters out of my car and pass them along each time he picked them up. After one visit, I caught him pulling them out of his trunk before walking them up to my door (the booster seats, not the kids)! He didn't have the guts to tell me he refused to use them, he just pretended, so I wouldn't nag. Later, I learned he let the kids ride in the back of his GMC Jimmy just like our parents used to do with station wagons in the 70's, before people used seat belts! I was so angry, then! But he and his wife just had their first child (so, his 2nd parenting experience). I cannot imagine for a minute that he would consider not using booster seats and seat belts for THIS kid!!