detailed rules when playing with the clock

Hello, I have a general question about chess rules when playing with the clock. Imagine the following situation:

white: Queen , King. But only 10-15 seconds left.

Black: King. Several minutes left.

So , in theory, won game for white..

White gives check (not mate yet) and stops his clock. Black moves his King and stops his clock. After a couple of seconds, white notifies that black has moved his King to an invalid square , where the King is check again. So white stops the clock and black moves the King back to the square he came from. 

Questions at this moment in the game:

-Has Black already lost because he made an illegal move? (And white has lost a couple of precious seconds)

The game continues...

Black moves his King on a valid square. And stops the clock. White moves his Queen and gives check, but..... before he can press the clock, his time is zero.

So, who has won this game?

If your opponent makes an illegal move in a classical time control game, then you can pause the clock and tell the arbiter. Generally you will be awarded a few extra minutes on your clock (even if you still have plenty of time). If your opponent's behavior is bad and he keeps doing things like this over and over he can be forfeited.

If it's blitz, then different rules are used. Sometimes it's an immediate loss for the player leaving their king in check.

In FIDE, if you run out of time, but there is no series of legal moves where your opponent can deliver checkmate (e.g. they only have a king) then you don't lose, it's a draw.

In most online sites like chess.com, if you run out of time, but your opponent only has a king, then it's a draw.

That depends on whose ruleset you're using.  I'm not fully versed on the FIDE rules in this regard, so I'll make a note of some generalities until someone more knowledgeable comes along.

First of all, when someone makes an illegal move and this is brought to the attention of the arbiter, then they typically will punish that player by giving their opponent extra time.  So, white catching black with an illegal move in your example might give them the extra time they need to be able to finish the game.

Secondly, many tournaments these days require increments or delays on their time controls.  This would give the white player a few extra seconds every move and as long as they can move within that time, their time won't go down- living on the delay, so to speak.

Thirdly, there are conditions under which a player may stop their clock and claim a draw based on the fact their opponent is trying to win only by making you run out of time ("unable to win by normal means" I think is how the FIDE rules put it; "normal means" refers to the possibility of checkmating the opponent).  These conditions vary depending on the ruleset being used. 

If the white player in your example were comfortable with a draw, they could likely stop the clock before their move and claim a draw.  Typically, they must still have time on their clock to claim such a draw.  Although, since black has only a king, I think just about every ruleset would declare that a draw on white's running out of time.  Add a black pawn that is across the board and far from the promotion square, and that could be ruled a win for black on white's timeout, depending on the ruleset being used.

Thank you for your comments. I would have thought that the rules would have been more clear. First, this was not an "official" game. Just a home game with a friend. Clock at 30 minutes each. The situation is clearly a win for white, unless.... he runs out of time. I conclude from the remark

"In most online sites like chess.com, if you run out of time, but your opponent only has a king, then it's a draw."

the described situation would be declared as a draw. Ok. Thank you

This!  As a USCF Tournament Director I've had this problem and USCF rules allowed me to award extra time to the abused player.

So if I resume correctly the above:

If Black has only his king left, and white has King plus a couple of other pieces left , regardless if he has a clear won position, if he runs out of time it's a draw?

Or more general: When the white player runs out of time it's a draw instead of a win for the black player?

Yes

Ahh, I always thought that when a player runs out of time, the other one has won automatically...

Then you can use whatever rules you want. You can allow take backs, add time to the clock when it gets low, etc.

But if you want to use tournament rules, yes, they get technical. The USCF rulebook is over 300 pages for example.

It helps if you set the clock for increment or delay, that way an overwhelmingly won position can't be lost (or drawn) due to the clock (as long as the player is not a total beginner).