Let's say that a piece is protecting my king from check, i.e. blocked or pinned. For example, my rook is in the way of my opponent's queen in checking my king. Can I checkmate their king with another piece, let's say my queen, if my check-preventing rook protects my queen, even though I could technically not move that rook?
Yes- if they captured your queen, then it would count as them going into check:
(from this, ironically listed as a "Related" question here)
Besides making sense, it's also explicitly stated in the rules of the game:
3.9 The king is said to be 'in check' if it is attacked by one or more of the opponent's pieces, even if such pieces are constrained from moving to that square because they would then leave or place their own king in check. No piece can be moved that will either expose the king of the same colour to check or leave that king in check.
Source: Laws of Chess on the FIDE site