Questions about pawn moves in Lasker vs Capablanca game

I'm a beginner and I would like some help, please.  Following is a game between Lasker and Capablanca (1915) http://www.chessgames.com/perl/chessgame?gid=1316949

(I'm having problems copying and pasting the game.  Sorry!!)

If you were kind enough to look at the game, I have some questions on the game. 

1. In move #35, why did the white king leave the c4 pawn and move up to e4? 

2. In move #36, why did the black king not take the c pawn and move to d6? 

3. Could Lasker have won this game and if so, how? 

Thank you very much in advance! :)

For your question 1 and 2 see above game overview with notes.

Question 3. : The pawn ending was worse for Lasker.

The shattered isolated pawn majority for white on the queenside could only have become dangerous when the black king had been too far away, and even then a pawn sac would have been needed to create a passed pawn giving black also a passed pawn there, and precise counting would have been needed.

The nice kingside majority from black as well as the weak white c pawn however tied the white king to stick around in the center.

The black king was active and could freely attack the white c pawn as well as defend the black pawns when needed.

Thank you so much, Achja!  This is very helpful :)  This will be my study material today :) 

  Please keep in mind my underwhelming rating when you read this, TAS.

...32  As Capa has said, to queen a pawn, start with the passed pawn.

33.  Lasker tries to stop the reinforcing pawn that will be needed to push the passed pawn through.

...33. Bringing up the reinforcing pawn needed for the passed pawn.  Notice how if he had tried to hurry things up by playing  g6 instead, with the intention of then playing f5 the next move, then Lasker would have played g5, thereby holding back both, the f & g pawn!

34 fighting to hold off the reinforcing pawns all the way down the line.

...34 the reinforcement to the reinforcement pawn.

 Eventually the passed pawn sacrifices itself and the 'reinforcement' pawn takes its place to go on to be queen.  I just think this game shows very cleary how pushing a passed pawn works in general.

 In anwer to your first two questions, I'll refer back to the notes in the game.  "Ke4. With the idea g4g5! 35.Ke4 also prevents f6f5 for black."

 If the Black King isn't there to protect the e pawn when White plays g4 to g5, then Black would have to play f5 and lose the passed e pawn or take the g pawn allowing white to take with the h pawn which would then hold back Black's h pawn.  All in all, messing up the smooth operation of pushing a passed pawn through.

Thank you, Joseph :) I'll keep your comment in mind when I look at this game again over the next few days :)