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No, castling can only be done with the rook.  The king is moved two spaces toward the rook, and the rook is then placed on the other side of the king.  There are other limitations as well, all detailed in the rules.

If it is black's turn, his king is in check, and he has no legal moves, then it is checkmate and he loses the game.  You can never have a stalemate while you are in check.

I can only assume that either the computer you played with had a bug, or you aren't remembering the details of that situation correctly.

Alright. So, here's my scenario: I played a game where the Black King was alone, against a White King, White Queen & a White Pawn. The Black King avoids moving into check. Meanwhile, the the White Pawn gets promoted to White Queen (2). Once this happens, that piece (i.e. White Queen (2)) is then putting the Black King in check (and being guarded by White Queen (1)). So, it is the Black player's turn but he has no legal moves & is now in check...
I'm assuming this means that he IS in check mate, but I've seen this maneuver on the computer-ages ago-& it ruled this a stalemate. I can kinda see the argument that it IS the Black player's turn & he has no legal move. However, since he IS in check, I'd think this is a checkmate... Which is correct?

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.

Thanks for the info.

EDIT:
Anyway, found the answer for my own question

From f.i.d.e.

'5.1
The game is won by the player who has checkmated his opponent’s king. This immediately ends the game, provided that the move producing the checkmate position was a legal move.'

'....This IMMEDIATELY ends the game.'

This means the other player checkmates him and he lost, and not that he WILL lose in a turn that where he is checkmated because he will not be able to get out of it and then player will capture him.

The FIDE rules cover situations that can occur in FIDE Chess. If a variant allows new situations to arise (such as having multiple Kings, or two pieces in the same square, or Pawns on the first rank), it has to specify how to resolve them; they are not covered simply by saying 'FIDE rules apply'. (Ralph Betza once attempted to codify a few rules that were used often in variants as 'Rule Zero', but while he undoubtedly used those rules a lot, I'm not sure whether they're any more common in general than other options, and in any case are not extensive enough to save very much repeating.)

If you're trying to discern the 'spirit' of the rule, I believe it came about something like this:

1. Initially, the goal of the game was to capture the enemy King, and 'check' didn't matter.
2. People got annoyed when an interesting game ended prematurely because one player made a dumb mistake that allowed his King to immediately be captured, so they decided to prevent that by making it illegal. Thus, if you make a move that would cause you to lose on the VERY next turn, you must take it back and do something else (if you have any other choice).
3. To get the current FIDE rules, you need to add the additional rule that a player who is not in check but who cannot move without placing himself in check (that is, a player in 'stalemate') receives a draw instead of a loss. It's not obvious (at least to me) why this should be so, and historically various players have resolved stalemates in just about every different way you could imagine, but the modern accepted resolution is a draw.

So the 'spirit' of the rule (in my opinion) is 'the REAL goal of the game is to capture the enemy king, but as a safety net, you're not allowed to make any move that would allow your opponent to win on the very next move.' In a variant that nullifies this safety net and allows you to place yourself in check, the most natural rule would be that the game is won by capturing the King, and placing yourself in 'check' is generally a poor strategy but otherwise has no special significance.

I was talking about variants that follow 'all f.i.d.e rules except...', or when someone want to follow f.i.d.e. chess logics/rules/idea while he is creating his own variant.

Um. They're called 'variants' because they all have different rules. There's no one rule that all variants follow, on that or any other point.

Thanks for the answer.
Now I see that the situation of both players being in check at the same time is impossible to happen in f.i.d.e. chess.

But, and for variants that because of their rules changes, allow the situation of both players being in check to happen, what would happen in this situation?
A player that becomes in check and cant get out of it would lose the game or he will only lose if the other player can capture his king?

Both players cannot be in check at the same time. The players must have missed that either party was in check and continued to play. They should go back to the position where the first check occurred and begin anew from there.
/Mats

When you are stupid enough to lose to an eleven year old, is it okay
to claim that he can not claim the win because he was stupid too and
said check instead of checkmate?

Re answer from 2007
'Cindy: the Black Queen cannot move since that would expose the Black King
to check, which is not allowed. This is a very useful tactic.'

The queen can move so long as it stays on the same row, between the rook
and the king.

Fergus writes: 'It is possible to use repeated checks to force the same position to repeat three times.'

See this page for a moving diagram of perpetual check. It is worth noting that sometimes the 50 moves rule will apply before the 3-times repetition rule. As I wrote back on [2004-08-27], in my database is the game [R. Pert - M. Franklin, 1996] in which both players have two rooks on the board. Black sets up a possible stalemate position on move 33 by advancing his passed Pawn to h3. All White needs to do is sacrifice both Rooks. After 21 consecutive Rook checks, they agreed to a draw. Of course, Black can always end the checks (and stalemate White) by capturing the last Rook.