Holy freakin' energy batman!

[mom-in-alberta]

You poor girl. I feel your exhaustion!!!
For one; I think you are right that mom and dad have not enforced any boundaries or anything like that. I would absolutely talk to them about the safety issues of having him running amok with smaller people around (in different wording, of course!). They need to begin to teach him to be gentle/respectful with others.
I wonder; most kids do start out quiet, and get more comfortable as time passes. Maybe he is going the other direction though? As in, he is sooooo overstimulated and excited to be there that his energy level ramps up? Hopefully, if that's the case, he will gradually settle in. Not to mention, he will pick up on YOUR house rules and learn that what he does at home won't be tolerated.
I think you have a good start in limiting what he has access to. Seems mean, but it's not. When he learns to play properly, he can run free like the others.
It makes me a little nervous that he laughs at others misfortunes. But empathy can be so hard to teach at that age. I would say "Please don't laugh. She's crying because she is hurt/owies/etc and that's not funny." Not sure you can do much else, though. If I had to guess, mom or dad have exposed him to tv shows, etc in which someone falling down, etc is funny. One of my friends used to watch Jackass with his kids (I know!!!) and they would laugh uproariously if someone wiped out.
To make a long story, well, long- I don't know that I would term until around the 2-3 week mark. If he is full time, you should begin seeing some improvement, at least. If nothing is changing, and you don't see mom and dad being on board.... bye bye!
On the plus side, if you are able to reign him in a little, you will have a kid who actually stays quiet during quiet time and has a relatively short day, to boot!! Not too shabby.
Good luck, and let us know which direction you go!!

PS- "I lub you so bad"... too cute.

[Dreamalittledream]

One of my friends used to watch Jackass with his kids (I know!!!) and they would laugh uproariously if someone wiped out

Speechless!! Wow!