We are currently VERY well-versed in the less-than-ideal neighbors. It's a couple with twin boys a year older than our oldest. We thought it was going to be great for our son to have 2 boys so close to his age to play with and help with the transition (he was very reluctant about moving out of our old house that he had lived in for almost 5 years).
It has turned out to be a nightmare from the start. Our very first night in the house, she comes over at 8:15 and says, "I know it's your first night in the house and all, but our kids go to bed at 8, and I can't keep my kids in bed because they keep hearing your kids." Say WHAT? I'm sure you can see the reaction that induced. However, I was very pleasant and apologize for the noise and explained that bedtime for us was usually around 8 as well.
Already put off the neighbors (well, her, anyway), a few days later, our kids were all playing in the common courtyard (it's a duplex), and one of their kids kicked my son. No reprimand, no words, nothing from the parents. She just came out and told the boys it was time to come in.
Believe it or not, a few months after we moved in, I was a meeting, and she came over to complain at 8:15 again. My husband said, "Yeah, we hear you, too" and slammed the door in her face. Ever since then, she won't look at us and actually scurries into her house and out of our way whenever she sees us. We actually think it's funny and aren't offended at all.
Simply put, she's crazy. He's pleasant, and their kids are never outside, so it's not too much to worry about. Many things have happened over the last 18 months.
However, in a situation like yours, I would absolutely have gone to the neighbors in a pleasant, unconfrontational way to address the issue right from the start. Having gotten no response, I would've indeed taken pictures and sent them to the landlord and kept on top of it until it was taken care of.
In no way would I make my kids clean up ANYthing that's their responsibility, except maybe some garbage because it helps them see that it's important to do their part to keep things clean (of garbage). However, I think you should immediately stop anything else because honestly, it would breed resentment in me, so I can't imagine how my kids would feel. With my luck, my kids would start throwing their stuff in frustration (haha).
Address it with the landlord, stop cleaning up after them, stop making snide remarks, and ignore them.