What a yucky situation to be in! So sorry!
Not sure I have much advice to give, but a couple ideas that worked for me when it comes to nagging and misbehavior...
When I had young 'monster' visitors (in my case, it was a niece and nephew) and the nagging started, I'd say, "That's nice". They weren't sure at all what to do with that answer.
"I'm thiiiiiirsty!" or "I want a drink!"
"That's nice" (said in a tone like- nice to know!)
And at first, they won't know what you mean. I'd wait and then prompt them (how can you fix that?) and give them that answer- "ask me this way" and give them the sentence you want them to use.
The next time they nag and you say "That's nice" they get the light bulb moment on their own and it's like they solved their own problem (using your solution).
(BTW, I used this with my gymnasts often! It's amazing how well it works! You're not giving them any kind of answer except that it's nice to know. They are left standing there trying to figure out what you meant. So they have to find another way to 'get their way'. I fell onto this idea when I was frustrated over my own kids demanding and whining and I got tired of reminding them how to ask for stuff because that method clearly wasn't working, lol)
As for the misbehavior, wow, that's a hard one. I've got one thing that comes to mind from my experience and it's this: my niece loved coming to my house because I had a cat she absolutely loved. This cat liked her well enough and it was fun to watch them play. But my niece had not been taught how to be respectful toward animals. And while yes, animals have to be trustworthy enough to take what toddlers and kids dole out, little humans have to be taught how to treat them. No tail pulling. No fur pulling, no riding or sitting on, etc. If my niece started to do these things, I ended their play. I set that boundary and followed through with the consequence. If they were spiteful, the consequence was a timeout away from everyone. If it was out of excitement and/or overstimulation, I'd just let them know that playtime with kitty was done and direct them to another activity.
In the beginning, I did have to constantly supervise but by laying out my rules every time they came to my house (KISS method here, especially for 5 year olds- maybe 3 general rules), things did get better.
Upon entering my home, we'd go through the rules. Then the learning curve where they figure out what those rules mean- which took a lot of watching, reminding, and yes, some moments with unhappy children dealing with consequences.
That way, even if that behavior is okay for them in their homes, they learn that it is not acceptable in your home. They can- and will- learn what's expected and accepted in differing environments, and lots of times they do that by trial and error. Do like school and lay it out for them.
Honestly, when it comes to physically dangerous things (like the kitty litter and food instance), I'm not sure what I would've done.
I did, however, have a young gymnast (I think she was 8 at the time and a regular in the gym) once who put her fellow athletes in danger by 'helping' too soon- she wanted to be so helpful and started to loosen equipment (the uneven bars securing bolts in the floor) before we were ready and if I hadn't caught it out of the corner of my eye, it could've been a horrible situation... she got an immediate reprimand and was not allowed to participate for the rest of that rotation. She felt so bad for her mistake, but also, I couldn't have that happen, even by accident. If that equipment came down on someone, it could've killed them.
There are some things that just can't happen... I may have ended the playdate right then and there, if faced with that same situation you found yourself in. But it's hard to say.
Damaging and destroying other people's property is a lesson that kids learn fast is a no-no in my home and it was the same in my gymnastics room. Unfortunately, if I had athletes or visitors to my home that didn't accept that, they were no longer welcome in the gymnastics room or my home. And it was not fun to enforce that rule, but I have had to do it on occasion.