I want to put the p-trap for my bathroom sink in the wall behind the sink, so that I have as much usable space as possible under the sink.

1. Will this be okay, is there a maximum distance that the p-trap should be from the sink?

2. Also, does it matter how low or high the p-trap is?

Are you thinking of sealing the wall so the trap is not accessible? It needs to be.

No, I will leave the trap open (i.e. cut out the dry wall in this area) and have the back of the vanity open in this area, so I can reach the trap.

If your stack in the wall is existing the Tee that accepts your lateral drain pipe is sticking staright out toward the front of the vanity. You would need to turn this Tee to be parallel with the wall surface and that means a 90 elbow or several 45s. I think you are doing a lot to gain very little.

So there is nothing that says that the p trap has to be right below the sink?

If I move the p trap to the wall and have the pipe coming to the sink (from the p trap) as high as possible, will the sink still drain okay?

It is a bit of extra effort, but I want to put a laundry basket below the sink.

Hewashuck... Just my opinion, and I really can't be sure, of course, but I might cut the basket so it slides past the trap instead of rearranging the plumbing... if possible. I just don't like the idea of moving that trap from under the tailpiece of the pop-up assembly. 'Nuff said. Good luck.

Yes and no, every time you add a 90 elbow to a drain you increase the chances of it clogging, you'd likely need 2 or 3 90s to get inside the wall and I think you'll have problems with it draining properly. You can shorten the tail piece from the sink drain only so much and then add a 90 to the wall, another 90 to the trap and yet a 3rd 90 and possibly a lateral to reacg the Tee on the drain stack. Your question about about this being OK also has some building code slant that I'm not sure of so maybe Tom or Growler will read this and reply too. If asked to do this installation myself I would decline. Hope Tom can help better.

If I move the p trap to the wall and have the pipe coming to the sink (from the p trap) as high as possible, will the sink still drain okay?
You will have built yourself a running trap with five bends, ( 3 90's and 2 45's)'in the drain before it ever gets to the vent stack. NOT GOOD PLUMBING PRACTICE! Since lavatories, tubs and showers have a lot of hair matted with grease in their drain lines be prepared for future blockages. I like Massplumbers idea about cutting the basket. Do it his way and save yourself a load of problems. Regards, Tom