My Mate in 3 puzzles
You forgot to remove the 2nd white queen from d1 generating an extra solution to your puzzle.
The White Queen on d1 cannot help Mate in 3. If it's moved, Black Nc6xQb8 and Mate in 3 is missed. The best solution I could find involving Qd1 is Mate in 5.

For the first puzzle, 3.Qh3 is also a checkmate option, but the equipped option leaves black without that rook XD
For the first puzzle, 3.Qh3 is also a checkmate option, but the equipped option leaves black without that rook XD
Are you referring to puzzle #1? If so, I don't follow you. Please clarify.
Mate in 3. Don't move the same piece twice.