I'm fairly skilled with Bobs, so I can help you here
First however, are you going to use just two Bobs, one for each player? If so it might be easier to simply use two sprites instead. Anyway, if you are going to paste something under a Bob, this is the way to do it:
Bob Update Off : Rem Turn off automatic Bob drawing
Do
X=X Screen(X Mouse) : Y=Y Screen(Y Mouse)
Bob 0,X,Y,1
Wait Vbl
Bob Clear
'All drawing must be done BETWEEN Bob Clear and Bob Draw
If Mouse Click Then Paste Bob X,Y,2
Bob Draw
Loop
Does "screen is refreshed or buffering" refer to Double Buffer or something else? Either may be problematic but it's hard to tell without more info.
So for the second request; to remove a pasted bob is a bit of a bother unless it was pasted on a solid colour background. In that case it's a simple matter of using Bar to remove it, otherwise you will have to save the background somehow before you paste the bob. One way to do this is with Get Block. Another way is actually to use Bobs instead of Paste Bob, because Bobs automatically saves the background underneath. That's the easiest way to go in my opinion.
Screen to Front switches between different opened screens.