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.
The king can capture any other piece in chess as long as capturing does not put him in check or checkmate. Wherever the king can legally move, the king can also capture.
You cannot take your own piece in chess because the rules prevent you from moving any one piece to a square that is already occupied by a piece of the same color. That means that a chess piece can never capture, or remove, another piece of the same color.
You cannot skip a move in chess as it is against the rules. What you can do is to make a stalling move to preserve your position. This is any move that doesn’t advance your game but doesn’t damage your position either.
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.
You can win at chess without losing a piece of your own. You can win a game of chess without taking any of your opponent’s pieces too. A most famous example is the fool’s mate which every beginner enthusiastically tries in their first games.
The king can definitely capture the queen in chess, though it is not easy. While the king may not be able to move very far in any direction as the queen can, it can certainly take a piece of any nature in any of those directions as long as he doesn’t put himself in check.
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.