Why is the F7 square considered weak?

Wow, really shows how sharp this kind of Sicilian can be... I presume that the idea was that if he goes Kf8, you will get a queen back with Ne6+, and probably some more after that.

Yes, if 15. ... Kf8 then White continues to operate with Knight checks and discovered checks even after recovering the Queen. One of the Black Rooks will drop as well.

Maybe simply move the king to h8 and try to unclog the mess black created with his pieces?

In reply to 14. Qxf7+?

After 14. ... Kh8 White just plays 15. Ne6 forking the Queen and the mate on g7.

Yeah I play Najdorf as black when I get the chance (I know this is Scheveningen so a bit different and white can try with g5 here) but it is a similar position. Luckily for me, up to this level, players with white rarely know how to go for the throat from the start. I often get same side castles and more positional games when I get Najdorf positions. And many players still goes either for not so good Bowdler attack or even some bad Smith Morra where queen takes the pawn and we get some weaker form of Scandinavian in a sense, where I keep both central pawns. 

Of course I get crushed from time to time in the opening (not that often). I go for this because I get many different positions as black which is not bad I guess.  

One question... if I still want to play the Sicilian which I do, would you recommend some other variation for black for me to try that gives somewhat of a more solid position, a bit less sharp than Najdorf?

I was playing the Najdorf from the time I was rated 1400 or so (back in the Late Bronze Age). Naturally, I occasionally got my head handed to me, but I learned a lot and became pretty good at defending and counter-attacking in that sort of position.

Have you seen my "Immortal game"?

A Heroic Defense in the Sicilian Najdorf - Kids, don't try this at home! - Chess Forums - Chess.com

Other variations? I suppose the 2. ... e6 and 2. ... Nc6 lines aren't as critical as the Najdorf, but also (in my view) not as exciting.

I do not see the fork,  but I do see the mate threat and queen attacked.

A fork is when one piece or Pawn attacks two vulnerable targets. In this case, the Knight on e6 attacking both the Queen on d8 and the Pawn on g7 (supporting a mate threat).

Yeah, I figured as much. 

I saw the game now.  Yeah that was insane. Defending under pressure, it's pretty hard, one of the toughest things to do in chess. I managed to win a few games where my opponent sacrificed a piece (and a sac was at least possible and not unsound) , but it was very difficult for me to win those games. I think that in most of those few I had at least 1 game losing blunder which wasn't utilized. As humans we just don't like to be on the defensive.