Very basic, but a good lesson for a beginning player. Nice! :)
Chess Lessons
ShiViChess, you probably have good intentions, I suggest putting in some time to try to grasp these diagrams yourself before you post them as lessons.
ShiViChess> Lesson #3: { Hope seems lost. Black could easily play Rh1#. How can white keep from losing?}
Always look at checks and captures. White has at least four saves:
1.Qd4+ Kg8 2.Ra8+ (2.Qxf2 also wins easily) Re8 3.Rxe8#
(1...f6 2.Qxf6+ Kg8 3.Qg7#)
(1...Re5 is hopeless)
1.Ra8+ Kg7 2.Qg5#
(1...Re8 is hopeless)
1.Qxf3 (goodbye Rh1#)
1.Nxf7+ (the trickiest to play) Kg8 2.Nh6+ with mate to follow.
(1...Kg7 2.Nh8+ Kxh8 3.Qd8 Re8 4.Qxe8#)
(2...Kh6 3.Rc6+ Kg7 4.Qd7+ Kxh8 5.Ra8+ Re8 6.Rxe8#)
ShiViChess> 2. Ne5+ { The knight is moved to safety, and uncovers a check. }
Q+R form an easy mating net. The knight's safety is irrelevant. A win is a win no matter how few or many pieces are left.
ShiViChess> 4. Qg6+ { Yet another check. The queen cannot be taken because of the knight. } 4... hxg6 { The only legal move is hxg5. } 5. Nxf3+ { The knight forks the rook and the king. If the king is attacked, the king can move. }
The queen sac serves no purpose. You could've immediately played 4.Nxf3 and won his rook without sac'ing your queen.
ShiViChess> 17. Nxg5 { The knight is gone. White has virtually won. }
White was won in the initial position. Ok, let's look at the positives. You were able to convert a queen-up ending in 17 moves. That's actually a great start. But next time, try to consider all CHECKS and CAPTURES. In many cases you found a move that didn't toss the win, but wasn't nearly the fastest or easiest win.
Puzzles #1 and #2 are fine. I finally switched to the move list on #3. 17 moves? As pointed out, Ra8+ wins in three. Qd4+ wins in four if Black throws the Rook and Pawn as delaying sacrifices.
Always win as fast as possible. A complex 17 move combination is in no way sound if mate in three is on the board. You give yourself too many chances to slip up and throw it away.
Presumbably Black will resign after Ra8+ (best) or Qd4+ (next best, and the first combination that I looked at:)), because they will suddenly see that it is over.

Lesson 1