Partial Answer
A Regular Rook
1. h4, h5
2. g3, g6
3. Bh3, Bh6
4. Bg4, Bg5
5. hxg5, hxg4
6. Nf3, Nf6
7. Nd4, Nd5
8. Rxh8#
Optimal, clarified by OP.
A Promoted Rook/Promoted Queen
1. h4, h5
2. g4, g5
3. hxg5, hxg4
4. g6, g3
5. Bh3, Bh6
6. Be6, Be3
7. Nf3, Nf6
8. Nd4, Nd5
9. g7, g2
10. gxh8R# for the Rook case.
or
10. gxh8Q# for the Queen case
Not optimal.
A Regular Bishop
1. g3, g6
2. Nf3, Nf6
3. Ng5, Ng4
4. Nc3, Nc6
5. Nd5, Nd4
6. Nf4, Nf5
7. Nxg6, Nxg3
8. Bg2, Bg7
9. Bd5, Bd4
10. Bxf7#
Not optimal.
A Regular Knight
1. e3, e6
2. Ne2, Ne7
3. g3, g6
4. Nbc3, Nbc6
5. Ne4, Ne5
6. Nf6#
Optimal, clarified by OP.