Yes, you can checkmate with only pawns and there are two different ways to do so. The first is promote pawns to queens who can easily deliver a checkmate. The second way is to deliver checkmate by trapping the opponent’s king and delivering checkmate with a pawn.
There is no official rule governing who gets the white pieces in chess. In informal games, the traditional way to decide who is white is to conceal a piece of each color, one in each hand, and offer your opponent a choice of hands. There may be a rule implemented at chess tournaments that requires a specific method of choosing though if there’s not, you are free to reach an agreement on pieces in any way you see fit.
Yes, most chess tournaments are timed. It is down to the individual tournament organizer in most cases as to how this will work, though FIDE official tournaments are governed by FIDE’s rules on timing for individual matches. In order to allow players to play at a very high standard, many tournaments may allow additional time based on number of moves played.
Any move can be repeated in chess, as long as that move is a legal move, and thus the position on the board does not prevent a move from being repeated. Although many would argue that this loses “tempi”, a move in chess can be repeated without any official forfeit. The only exception is when both players repeat their position three times over – in this case the chess game is drawn.