White to move. How should white proceed?
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
3.Qb3 Rf3 (leaves the queen pinned)
4.Qxf3 Kb1 (the king has a move)
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.
It's a winning position for white.
As for the third puzzle
Other moves create the threat of stalemate.