Puzzles from missed Tactics

Yes. Qg7 maintains Black’s advantage while the obvious move Qxh7 (which I played) results in a draw by repetition (I saw this, but didn’t see the more subtle and counter-intuitive Qg7).

Correct

Bg1+ is what I played in this game, but Bg1+ is not the winning move.
After Bg1+ the best Black can do is draw.

1...Bc7! 2.Qb4 Qf2+3.Kh1e4! (Bb6)

Correct 

Хорошо.

I would go for Bxc5. After Rc8 Bxe3, Rxc2 Bxf4, gxf4 the endgame should be fine for black down the exchange but with a solid pawn structure.

Yes, that's correct. Black need not fear the pin because they can set up "a fortress": https://en.wikipedia.org/wiki/Fortress_(chess)

The rest of the game can be found below:

In this recent game, both sides are looking to develop their pieces to their best squares. White would like to develop their dark-square bishop to e3, but is reluctant to do so because Black can play f4, forking both bishop on e3 and knight on g3 (as well as the fact that moving the DSB allows Black to play Qxb2). Should White be afraid of this pawn fork? Why or why not?

I definitely don't have anything like a full line, but certainly after Be3 f4, white has the e4 square for a knight to hop into and the black king is extremely weak - for instance combinations like Bxf4 and then Re1+ or Qe2+ look strong, and so does allowing fxe3 fxe3 opening up the f1 rook. 
It looks like white might have a massive positional advantage despite the loss of material, but I can't see any clear wins - maybe others can?

One of the great things about learning 'advanced tactics' is that you begin to see possibilities in a position that a lesser player (which was me in this case) fail to see.
@Iron_Helmet. You are on the right track when you recognise that Black's king is slightly vulnerable sitting on the half-open e-file and can no longer castle. You are also right to recognise that for Black to play f4 would be dubious because it weakens the e4 square, allowing either knight to be able to jump into that central outpost. But it also weakens another square and this is key to the solution of this puzzle.
Unfortunately, playing Bxf4 in response to f4, neutralises Whites advantage and at best results in a draw. And Bxf4, exf4, Re1+ while looking promising, gives Black the advantage after the Black king runs to safety.

 Be3! f4 d4! fxe3 dxe5 Now Nd5 and Qe1 look good, as does f4(as the first move).

@fruitmonster. You are right that Be3 is best. White need not fear Black playing f4 but d4 is NOT the answer. You are right that if Black responses to d4 by playing fxe3?? than dxe5 gives White a winning advantage, BUT if Black plays 18...cxd5 (instead of 18...fxe3) Black is the one with the winning advantage.

I’m going to give some more time before providing the solution to my last and most difficult puzzle.

Интуитивно, сильно выглядит 2.b4, вскрывая линию "b", чтобы использовать неудачное расположение черных фигур и слабости в позиции черных.

I'm thinking about the royal fork Nd5+, but Bxd5 so that's out, if we don't have the d pawn in the way, Qd7+!, are these part of it??

In the last puzzle after 1. Be3 f4 I thought of 2.Qg4 fxe3 or fxg3 3. Qd7+ Kf6 4. fxg3 or fxe3 (disc. check) as main line