Yes, a pitcher can be moved to another position temporarily and then moved back to pitcher. Managers often made use of this strategy during the 1980's, moving a pitcher to a corner outfield position for one batter and then moving him back. As managers started carrying more and more relief pitchers on their rosters the need for this sort of maneuver lessened.
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Yes. MLB Rule 3.03 Comment states: "A pitcher may change to another position only once during the same inning; e.g. the pitcher will not be allowed to assume a position other than a pitcher more than once in the same inning."
Only the Pitcher and Catcher have defined positions, other defensive players can stand anywhere in fair territory (Rule 4.03c)
Yes, Only if he is put in a different position. He also must stay there until the next inning and then he can go back into the game and pitch.