Is it true that a good "flanker" (flanking position) is worth more than a pawn? If so, why? Perhaps I've used the wrong term, but I'll describe what a "flanker" is.

In the Queen's Gambit, there is often the possibility for White to sacrifice a bishop on h7.

1.Bxh7+ Kxh7 2.Ng5+ Kg8 3.Qh5 Kf8 (the rook is on e8) 4.Qh8+ Ke7 5.Qxg7 (taking the g pawn.) White has two pawns for a piece, is down a third "pawn" and usually has the advantage. Why is that?

Some observations:


1 White has a passed h pawn that is a queening threat in the end game, and possibly even in the middle game.
2 White dominates the g and h files, where Black pieces have no safe haven. It is this domination of two "edge" files that make for the "flanker." Black is confined to six files.
3 Derivative from b) White controls at least eight squares on Black's side of the board. If Black has nothing beyond the fourth rank on the a-f files, he is limited to 24 out of 64 squares.


This effect is so strong that it even extends to positions that I would classify as "quasi flanker".

For instance, in this "Immortal Zugzwang game, Black sacrificed a knight for two pawns so that his rooks on the f file act as a "flanker" against White's whole queen side: 

http://en.wikipedia.org/wiki/Immortal_Zugzwang_Game

[FEN ""]
1. d4 Nf6 2. c4 e6 3. Nf3 b6 4. g3 Bb7 5. Bg2 Be7 6. Nc3 O-O 7. O-O d5 8. Ne5 c6 9. cxd5?! cxd5 10. Bf4 a6 11. Rc1 b5 12. Qb3 Nc6 13. Nxc6 Bxc6 14. h3? Qd7 15. Kh2 Nh5 16. Bd2 f5! 17. Qd1 b4! 18. Nb1 Bb5 19. Rg1 Bd6 20. e4 fxe4! 21. Qxh5 Rxf2 22. Qg5 Raf8 23. Kh1 R8f5 24. Qe3 Bd3 25. Rce1 h6!! 0–1


And in this 1919 game between Capablanca and Yates, White sacrifices a rook for the light squared bishop and a pawn between moves 22-24, and comes away with a "quasi flanker" on the queen side (and a winning advantage).

http://www.chessgames.com/perl/chessgame?gid=1055968

[FEN ""]
    [FEN ""]
1. e4 e5 2. Nf3 Nc6 3. Bb5 a6 4. Ba4 Nf6 5. O-O Be7 6. Re1 b5 7. Bb3 d6 8. c3 Na5 9. Bc2 c5 10. d4 Qc7 11. Nbd2 Bg4 12. d5 g5 13. Nf1 h6 14. Ng3 Rd8 15. a4 b4 16. cxb4 cxb4 17. Bd3 Bc8 18. Be3 Ng4 19. Rc1 Qb8 20. Bd2 Qb6 21. Qe2 Nb3 22. Rc6 Qa5 23. Bxa6 Bd7 24. Bb5 Bxc6 25. Bxc6+ Kf8 26. Qc4 Nxd2 27. Nxd2 Qa7 28. Qe2 h5 29. Nf5 Bf6 30. Nc4 Qc5 31. b3 Nh6 32. Nxh6 Rxh6 33. Qe3 Rc8 34. Rc1 Bd8 35. Qxc5 dxc5 36. Nxe5 Ke7 37. Rxc5 f5 38. Rc4 Ba5 39. Bb5 Rxc4 40. Nxc4 Bc7 41. e5 Bb8 42. Ne3 Rh7 43. Nxf5+ Kf7 44. e6+ Kf6 45. e7 Rxe7 46. Nxe7 Kxe7 47. g3 Bc7 48. Kg2 Kd6 49. Be8 h4 50. Bf7 Ke5 51. Kh3 Bd8 52. Kg4 hxg3 53. fxg3 Kf6 54. Be6 Kg6 55. d6 Kf6 56. Bf5 Bb6 57. d7 Bd8 58. h4 gxh4 59. gxh4 Bc7 60. h5 Kg7 61. Be4 1-0


What compensation did the winners get for their sacrifices?
I will give you my view, though I'm only a FIDE 2200 player. Such rules do not apply in all games and you usually should not let them affect the game. That does not mean that all rules you hear are not valid, for example take the first example: a sacrifice of a piece for two pawns and attack on the king is usually good compensation. But it always depends on the position, it is up to the player to determine the strength of the attack. I would go with the known rules using them carefully. All rules have exceptions.

Compensation in the Nimzowitch game is perhaps the hardest to explain, I would guess that he saw the end position before the sacrifice since he was great tactical player.

In the Capablanca game the compensation is more clear, a bishop, a protected passed a-pawn and white square dominance and it is true that black is so badly coordinated that he cannot effectively use his kingside flank.

To the original question, it is always some advantage that compensates for a pawn and yes, it could be control on some part or side of the board. It is always up to you to determine the extent of the advantage. Your observation no.3, though: there are openings where black is confined to little space but gains a flexible position so space advantage alone is usually not compensation in itself.