Is this really checkmate?

If the white rook just moved up, is this really checkmate since white's queen is pinned? Would the rook have to be supported by a piece that can move since the queen cannot? Just a bizarre question!

Yes it is checkmate.

The black king is in check and can not move anywhere.  Therefore it is checkmate. 

Yup! Checkmate it is! The black king is in check and cannot capture the rook since the queen is protecting it!

Yes. The immediate threat has to be dealt with first. That's the beauty of that rule, if you use it to your advantage. Sometimes you can distract or move an enemy King away from another piece if that piece is already pinned.

I am sure any chess rule book, assuming this is in there would agree with you all but think about it for a second. The king takes the rook but how can the queen take the king if the queen is pinned? Common logic says it cannot.

Because as soon as the white queen takes the black king, the game is over, and the black queen would never get a chance to take the white king.

Since you are a fan of common logic, as you so well put it. Think of it this way then, imagine that you actually had to take the king with the pieces.

1... Kxb8 2. Qxb8. And the king is dead before your queen has the time to take your king, now we all know this isn´t how the rules are, but it can be used to show why a pinned piece can still threaten squares.

The rules state that the King can not take a piece that is protected by another--here White's Rook is protect by his Queen--the pin of the Queen is irrelevant. 

Yes, you might expect the king to be able to play Kxb8 because the white queen is pinned, but this is simply a chess rule.

Also, of course, a pinned piece can deliver checkmate, even though it can't actually capture the king.

3...Kd8? (...Kc8 would delay mate two moves I think) 4.Qa8#!

could be an interesting variation on the rules, king can only be checked by a piece that could potentially move to that square otherwise it is safe. Although in the first diagram black would still be lost as white would simply take black's queen after white's king takes the rook

very good question, I believe it is a checkmate. Thanks for asking this is is good to have to think about this type of check-mate

Reminds me of a timing play in baseball.

In the first diagram, it is impossible for it to be "white to move".  Just a note.

No it is not impossible for White to move in the first diagram--Besides putting Black in checkmate he can take Black's Queen--a totally idiotic post

It cannot BE white to move as black is in check (actually, checkmated).  Think before you post insults.  "White to move" means that it is white's turn, and the comment is meant to let the OP revise his post to say "black to move".

In the position given black is checkmated, game over, white wins.  nuff said 

The goal of the game is basically to take the enemy king before he takes yours. Since checkmate means that the king cannot escape, it is logical to stop the game at checkmate instead... but if the game continued to its logical conclusion (ignoring basic checking laws which are built on this premise), then Black could either play 1...Kxb8 and be taken by 2.Qxb8, or he could try 1...Qxb1 when he'd be taken by 2.Rxa8.

Think of it like this: whoever takes the king FIRST wins. Now play through the moves from your position: Black plays KxR, then White plays QxK -- that's it, the end, game over. If given one more move, Black could capture White's king and have revenge, but that's not the case -- his king is already dead and therefore the game ends IMMEDIATELY. Black is already "dead" and can no longer fight back.

If you are in a one-versus-one gun shootout, whoever gets shot first dies. You don't get to rise from the dead and shoot your opponent after he's already killed you FIRST.