checkmate rules


1) Yes, if you have taken all of somebody's pieces and you can't mate them in 50 moves, a draw may be claimed.
(It has to be claimed by pressing the draw button. It is not automatic.)
If the side with material remaining runs out of time, a draw will be declared because the side with the king only could never checkmate its opponent no matter how badly the other side played.
If both sides have nothing but kings left, the game is a draw.
2) If a pawn hasn't moved and there is nothing blocking it, it can jump one or two squares, unless that pawn is pinned and the move somehow opens its king to check.
However, there is a special rule called "En Passant" that may come into play here. If a pawn jumps two squares but a move of only one square would have left it subject to capture by an adjacent pawn, it can still be taken and the capturing pawn is placed on the square where it would have gone, had the pawn only gone one square. This special capture is allowed only on the very next move.
(Easier to show than to fully describe.) Search for "En Passant" captures. Wikipedia does a nice job of showing en passant captures.