Yes, H is neutral because the chrages do cancel out.
No, there's no positron in an H atom. If one was there, temporarily before it either annihilated or left, the resultant object (which would no longre be an H atom) would have positive charge.
You can make "positronium" in the laboratory -- it consists of an electron and a positron which are bound together because of their electrical attraction. It doesn't last too long -- electrons and positrons annihilate quickly.
You can also make "antihydrogen", which is an antiproton with a positron orbiting it. This atom acts, as far as we can tell, just like a hydrogen atom, except the charges are reversed.
I don't think there is a stable bound state of a positron, an electron, and a proton (I could be wrong on this, though).
(republished on 07/13/06)