Yes. It is common practice for a goal keeper to take a goal kick.
A goal may be a scored directly from a goal keeper punt.
general knowledge goal kick goal keeper
A goal may be scored directly from a goal kick, but only against the opponents. An own-goal cannot be scored directly from a goal kick.
most of the time when a goal keeper takes the penalty shot but sometimes when the goal keeper of the other team is so much above his field or area the goal keeper may kick the ball from goal to the other goal.
Goal kick. Although he may kick it from his hand whens caught in active play.
A goal keeper may take a corner kick.
Only if the keeper intends to place it in another location and take the goal kick. If the referee determines this to be tactic in delaying the restart of play, then the kicker is risking a caution.
If an opposition player pushes the goal keeper it is considered a foul and a direct free kick is awarded for the keeper's team. This is true for any player and not just the goal keeper.
If the ball crosses the goal line on it's own momentum, then a goal is scored, even if the keeper touched it. If it is a penalty kick during the match itself, then the kicker may only kick it again if it has touched another player, such as the goal keeper. If it is a penalty kick in extended time (e.g. time expired after the penalty kick had already been called) or if during kicks from the penalty mark (a.k.a. shootout) then no player may touch it a second time, except the goal keeper.