Well done- that is the line that explains why 1.Rf8 is the shortest mate. The first time I saw this problem, I found 1.Rf8 by chance- it was the second try at a solution for me, and at first, I didn't see this particular line where the queen mates at d1. Here are the two references: https://www.youtube.com/watch?v=-mAAnRJxN6Y
http://web.chessdailynews.com/a-chess-masterpiece-puzzle/

1.Rf8 Rd1
2.g8=Q Rd3+
3.Qb3 Rf3 (leaves the queen pinned)
4.Qxf3 Kb1 (the king has a move)
5.Qd1#

The third puzzle, however, is still not quite complete in my opinion because you still haven't explained the precise reason 1.Rf8 leads to mate in 5 while 1.Re8 (or 1.Rc8, for example) can only lead to mate in 6 at best. In other words, you are missing the critical line in 1.Rf8. In 1.Re8 line, black checks along the 3rd rank, and when white blocks the check with the queen, black can refuse the capture, but keep the queen pinned- this leads to the mate in 6 that you give above, however, when white has played 1.Rf8, explain why this same maneuver can't ever lead to mate in 6, but leads to mate in 5 instead.

V4 is the key line, though it is unobvious transposition of puzzle #2. So that one is complete.

Yancey, here is the solution for the second puzzle.

V-4

1.h3+ Kh4
2.Kg2 e1=Q
3.Rd4+ Bxd4
4.Nxd4 Qe4+
5.Nf3+ Qxf3+
6.Kxf3
It's a winning position for white.

As for the third puzzle

1.Re8 Rd1
2.g8=Q Rd3+
3.Qb3 Re3
4.Qxe3 Kb1
5.Qd3+ Ka1
6.Re1#
Other moves create the threat of stalemate.

As for the other puzzle, you leave out the critical line where black queens the pawn, though I do realize you understood it was still won for white. This was the line Troitsky wanted one to find eventually.

In the third puzzle, consider this line: 1.Re8 Rd1 2.g8(Q) Rd3 3.Qb3. Now, white can't mate in 5 total- it takes 6 moves. I am asking you to describe why this happens with 1.Re8, but 1.Rf8 prevents black from extending the line to six moves. You will other moves at move 1 for the white rook run into the same problem.

Yancey, Here is the solution for the second puzzle

V-2

1.h3+ Kh4
2.Kg2 Ba7
3.Rd4+ Bxd4
4.Nxd4 e1=N+
5.Kf1 Kxh3
6.Kxe1 Kg4
It's a win.

V-3

1.h3+ Kf5
2.Nd4+ Bxd4
3.Re7 Bc3
4.a6 e1=Q+
5.Rxe1 Bxe1
6.a7
It's a win.

As for the third puzzle only Rf8 allows to pin the white queen, but this pin doesn't make sense. For example

1.Rf8 Rd1
2.g8=Q Rd3+
3.Qb3 any move along the third rank allows mate

Reference for Puzzle #1 http://web.chessdailynews.com/endgame-brain-challenge/

In puzzle #2, what about black's reply of 1. ....Kh4? That is the main line of the puzzle, in my opinion. In addition, in the line you give, what about 3. .....Be5?

In puzzle #3, 1.Rf8 is the right start, but the line you give doesn't distinguish it from all the other moves white might have made with the rook on the first move. In other words, similar mates arise after, for example, 1.Re8, but they are longer by one move. Why is 1.Rf8 special?

Yancey, Here is the solution for the first puzzle.

V-1

1.d6! Rb4
2.d7 Rb8
3.Ne6+ Kf5
4.d8=Q Rxd8
5.Nxd8 Kg4
It's a win.

V-2

1.d8 Kf6
2.Nd5+ Ke6
3.d7 Rh4+
4.Kg1 Kxd7
5.g3 Rh3
6.Kg2 Kd6
7.Nf4 Rxg3+
8.Kxg3 Ke5

V-3

1.d6 Rxe3
2.d7 Re1+
3.Kh2 Rd1
4.Ne6+ Kh4
5.d8=Q+ Rxd8
6.Nxd8 h5
It's a win.

Here is the solution for the second puzzle.

1.h3+ Kf5
2.Nd4+ Bxd4
3.Re7 Bc3
4.a6 e1=Q+
5.Rxe1 Bxe1
6.a7 Bc3
7.a8=Q
It's a win

Here is the solution for the third puzzle.

1.Rf8! Rc1
2.g8=Q Rc3+
3.Qb3 Rxb3+
4.Kxb3 Kb1
5.Rf1#

Puzzles for Alena:
8/2N5/7p/3P2k1/4r3/4P3/6P1/7K w – – 0 0 White to move and win.
8/3R4/4N3/P1b4p/6k1/8/4pP1P/6K1 w - - 0 1 White to move and win. Troitsky puzzle
6R1/6P1/8/8/8/K7/6P1/kr6 w – – 0 1 Find the shortest mate.

@Arein22 Oh, OK

@DaveCubby I have give *1. ... Qg1+

@Arein22 1...Qg7 is illegal because Black g-pawn is in the g7 square.

*1. ... Qg1+

In every variation white is winning and it results in mate in 2 except the 1. ... Qg7 variation.

1. e6!

a) 1...Qg1+, 2. Rxg1 g6, 3. Qh8#
b) 1...g6, 2. Qh7# (not 2. Qh8+?? Qxh8)
c) 1...Qf6, 2. Qh8#

E6 and black cannot escape the Queen and Rook battery.