Pentomino tiling on wrap-around 5x5 grids

It is known that P pentominoes cannot tile a 5×5 square board.

Q1: If the east and west edges of the 5×5 square board are "wrapping around" (if you move a piece through one of the edges, the part appears on the opposite edge), can you tile it with P pentominoes?

Q2: In addition to the above, the north and south edges are also wrapping around (thus forming a toroidal surface). Can you tile it with P pentominoes?

Q3: Identify all distinct pentominoes that can tile the 5×5 toroidal board.