First let's try not to be judgmental. We do not live in a perfect world and unfortunately this happens. You/may or may not know what kind of situation the mother is in mentally and/or health wise. I know it can be a big pain and very irritating, but the good thing is that they are still young (not teenagers) so they are still impressionable and can be taught respect and values with the right coaxing.
I too was in a similar situation I actually moved next to the problem that was already there not knowing there was a problem. I have a single mother with a unsupervised 17 year old daughter, oh my, we had loud music, illegal use of drugs, beer bottles in the yard, condoms in the yard, vehicles coming and going, hungry continuous barking dogs, destructive dogs to personal property and trash, it was a rough year.
However with perseverance they changed their routines. I made several phone calls to the police and animal control. Parties which meant me standing outside until 3am monitoring their activities b/c of course momma was not home to supervise. I spoke with the mom (on the one occasion when she was actually at home) and those only by chance b/c the dog had came in my backyard and strolled trash all over it. That was the only should I say pleasant conversation we had. Finally at (about 7 or 8 months)I had to write a very detailed 3 page letter to their landlord and that seem to be the answer a lot of things leveled out after that.
I would like to recommend that you befriend the mother first, speak to her and communicate with her on her level and not appear to be belittling her, she sounds like she may get very offensive really easily. Also, allow your 6 year old to befriend the oldest, let them socialize inside w/close supervision so you can monitor the conversation and activities. Kids learn by what the see. The younger ones I would limit access b/c they pick up things really easily and it's hard to break bad/improper behavior.
The little one that's outside urinating it sounds like he wants some attention so that next time he does that and see him. Grab a snack/treat and go out to the street and ask him why does he do that and offer him a reward if he promises not to do that anymore you will reward him with a treat at the end of each week. I know you really shouldn't have to be doing this but sometimes you have to go out of your boundaries for the benefit of the whole team.
I agree to invite them to church even if Mom chooses not to go and if you can convince the kidz to promise to behave to take them church they probably need to get involved in some activities away from home and build some social skills.
If this is just too much for you to handle and if you know of someone that you think will be able to handle the matter better ask them for help; if you do attempt to intervene and it’s not welcomes and there is no/limited change then you have acknowledge the fact that you did try b/c sometimes with all that we do some people refuse to change, however always pray, for them, for you, for your kids, and ask God for guidance and strength.
God Bless Hopes this Helps