Dia. 1. Correct
Dia. 1. The correct endgame sequence
The hane of Black 1 is the correct move. White has no choice but to answer with 2. Black's territory in the upper right corner is now secure, so Black switches to the bottom left corner and connects at 3.

White ataries with 4, but Black ignores it and blocks with 5, forcing White to defend at 6. Black connects at 7, ending the game.

The result is a draw.
Black: 15 points. White: 15 points.

Dia. 2.
Dia. 2. Short of liberties
White cannot answer Black 1 with the atari of 2. Black will descend to 3. White is now short of liberties: he can't atari at 'a' or at 'b', so all of his stones at the top are dead.

Dia. 3
Dia. 3. A seki
Suppose White answers Black 5 in Dia. 1 by capturing at 1. Black will atari with 2, then attach at 4, threatening to throw in a stone at 5. White is forced to connect at 5. Black now plays a hane with 6. The result is a seki.

If White tries to start a ko with 7, Black will connect at 8. After 9, White will have to fill a liberty to start a ko, but Black will take the ko first and White has no ko threats.

Recommended reading
Get Strong at the Endgame is one of the most practical books ever written on the endgame. In it you will find the calculations for the value of endgame moves in over 100 common positions, as well as 120 endgame tesuji problems. Also included in its problems are 28 endgame positions on 11x11-size boards, where you can study the interplay between sente and gote. It may be that the biggest move is not the move that should be played first.