Okay, I look at deadbeat differently. I think of deadbeat as someone who is totally uninvolved with their kids. No money, no contact, etc.
I think he needs to man-up and take full responsibility for his kids in the time that his are with him. Meaning he needs to arrange appropriate child care if he has to work, and he needs to provide adequate housing for his children. I can only go by what the OP has posted, and that is that he lives in an apartment that is not suitable for children, and that he frequently has to work on the days he is supposed to have the kids. That's it.
I think that it is unfair for either parent to expect the OP to babysit. I don't think it is right for the daughter in law to be angry with the OP because she has another obligation. I understand that she is probably frustrated with the ex husband and took it out on the mother. I understand why she is frustrated.
As for child support, it is a formula. It is a formula based on the non custodial parents income, and the amount of time that the child spends with the non custodial parent. If the child spends 50% of their time with both parents, then there is no child support, because it is assumed that the parents each pay for 50% of the child's living expenses. If the child is with one parent in a more disproportionate amount of time, that parent receives child support. As it stands these children are with their mother 6 days out of the week. The child support that would be paid would reflect that the mother is responsible for the majority of their living expenses. Just think of it simply. The way it is working now, she feeds the kids 18 meals a week, he feeds them 3.
Now, granted, none of us know the true details of what he is contributing to the mother financially, and whether, or not, that is a "great" income or " a lot" of money is subjective. What I think is a lot, someone else might think is a pittance. But, having seen this 1st hand, and knowing that, often, courts are biased against men, things could get much worse for the son if the ex took him to court.
Also, kids are only little for so long. There is a very limited amount of years where they actually do want to hang out with their parents. Foster that bond now, establish that relationship now. Once they are teens they won't want to go to dad's or grandma's every weekend.