How can 2 pawns outrun a rook?

Black is winning in both those positions:

In your first example, White wins, no matter which pawn he pushes. Those pawns are connected and too far advanced to be stopped. (1. Kd1 is pointless when 1. c7 or 1. d7 win quickly.)

In your second example Black is winning until his pointless tempo loss of 31...Rc8?? [corrected]. White needs to move in another piece, however, especially his King, to help prevent those pawns from promoting. If Black's king weren't on the back rank then Black could hold out longer with a discovered attack with check, which would win a promoted queen with his bishop, then win the endgame.

You're right, I meant to say that in the first example white wins given 1: Kd1. However, in the second example, I showed that even after 32: Rd8, black still wins with Ba6 and giving up the rook for both the pawns.

1. Kd1 loses after either 1...Kf8 or 1...Kf7 in the first example. Push a pawn instead.

The king is a strong fighting piece! Even from the f-file he can sneer at these impudent pawns.

In the first diagram white won only because the rook was on d8, and a pawn could advance to c7 WITH TEMPO, not allowing the king to approach. 

In that same diagram, if black plays 2...Rxd6 he wins simply - as on 3. c7, ...Rc6 will take control of that pawn as well.

After both wrong moves, 1. Kd1 and 1...Rd8 - the game should end in a draw assuming best play. For example:

2. d7! Kf7 3. Ke2! (he needs to move the king out of the d-file, so that his d-pawn isn't captured later by the rook with check) Ke7 3. c7 Kxd7! 4. cxd8=Q Kxd8 draw.

In the second example, black is up so much material, that he can win even after 31...Rc8 - provided that after the further 32. d7 Rd8 33. c6 he plays 33...Ba6!

Black will take control of the c8 square, and is easily winning after that. He can bring his king over and simply collect all of the pawns - for example 34. Kf4 Kf8 35. Ke5 Ke7 and 36...Rxd7 - with an easily winning endgame.

whenever white plays c7, black can take on d7.

Black may even consider a move ...Bb7!? at some point - threatening ...Bxc6 and forcing white to capture that pesky bishop. 

Example: 34. Kf4 Bb7!? 35. cb Kf8 36. Ke5 (or 36. c4, doesn't matter) Ke7, after which black will simply collect all of the pawns and win easily. 

This is, in a sense, more precise than bringing the king over, as we remain up a whole rook in this variation - but it's also a bit more fancy and less simple... and of course an extra bishop is quite enough in this ending. 

Oops - Xman720 already gave my last variation (except for the ...Bb7 idea, it is true)...

Great minds think alike!

Yea, I didn't consider Bb7 in that variation (although I did on move 36 instead of Rxd7, unfortunately it fails because it gives the c6 square to the king after cxb7.) because I didn't see the idea of just moving the king over, and of course because planning rook takes is completely winning anyways, but good find.

<Xman720> Yeah, definitely a cute move, this ...Bb7 :-) !!!

<Morphysrevenges> still it's a bit surprising when you learn it for the first time.

Does it mean that the connected, passed pawn on the sixth is worth 2.5 points?

It may well be so, as often we see a minor piece sacrificed in order to stop it when it gets that far. 

It's nice to see how the prospect of becoming a queen influences the numerical value of a pawn as it approaches the eighth rank...

On a final note - the 1600-1700 player (and also much stronger players) would routinely miss combinations based on creating this situation, of winning (or drawing) 2p vs. R...