Is this a checkmate (or even check)? (Paradox-ish position)

The white king is in check and has no move to get out of it, so black wins.  It doesn't matter that technically the checking piece is pinned.

The last 2 diagrams are both checkmate. The White King has no legal moves and the White Queen can't capture the Bishop because it is pinned by the Black Rook in both positions.

I understand your query. Take the third diagram for example. Imagine it was Black's turn, and Black played Bxa2 (removing the king). Black's king would be exposed to the White queen and White could capture it, but the Black king disappeared already! After all, the objective of the game is to capture the enemy king before the enemy captures your king (simplied to simply 'checkmate' without really removing the kings from the board).

Of course it would be White to move in the diagram, but clearly Qxc4+ means Rxa2, so we simply mark Qxc4+ as an illegal move.

A simpler example is as follows.

If White plays Bc6+, yes the bishop is attacking the Black king, but in doing so the White king is also exposed to the Black rook, and with Black to move, of course Black will waste no time in capturing the White king first. Usually, we simply denote Bc6+ as an illegal move.

second diagram in the original is an illegal position... where did the black bishop giving check move from??

Second diagram is completely legal.