i'm wondering why black doesn't play c4 and b4 after b5 and c6 in this game. Wouldn't that be a tempo, i.e. the bishop on d3 had to move? Additionally it would gain black central and on the queens site room advantage, right? So why doesn't black play that?
If white plays
e4 with out black having played
...c4, then black can trade the central pawns with
cxd4 to reduce white's central control.
[StartFlipped "0"] [fen "r1bq1rk1/p2nbppp/1p2pn2/2pp4/3P1B2/2PBPN1P/PP1N1PP1/R2Q1RK1 b - - 0 1"] 1... Bb7 2. Re1 Rc8 3. e4 cxd4 4.cxd4 dxe4 5.Nxe4 Nxe4 6.Bxe4 Bxe4 7.Rxe4 Nf6
(Black can play 2...Ne4, which is stronger, but that's a different story.)
However, when black has played
e4 is especially strong because when black takes with
dxe4, white's increased control of the center is difficult to challenge, as black lost the option of
[StartFlipped "0"] [fen "r1bq1rk1/p2nbppp/1p2pn2/2pp4/3P1B2/2PBPN1P/PP1N1PP1/R2Q1RK1 b - - 1 1"] 1... c4 2. Bc2 b5 3. e4 dxe4 4. Nxe4
If black does not trade with
...dxe4 then white will play
e5 and gain an easy attack on the black king.
[StartFlipped "0"] [fen "r1bq1rk1/p2nbppp/1p2pn2/2pp4/3P1B2/2PBPN1P/PP1N1PP1/R2Q1RK1 b - - 1 1"] 1... c4 2. Bc2 b5 3. e4 Bb7 4. e5
Black can try to use his queenside space advantage and play for
b5-b4xc3 to target
c3, but this plan is slow and black is likely to get mated if he tries this.
From a positional viewpoint it is okay to keep the tension at the centre and instead of playing c4, Bb7 is preferable. The reason is that Black develops his bishop and lets White to wonder his/her next move. The position is quite equal and the loss of a tempo at this position doesn't give much
After Bb7, Black could play immediately a5 and try for an attack on the queenside.
The plan you propose doesn't seem to do anything since after
9... c4 10.Bc2 b5 11.e4 dxe4 12.Nxe4
Black doesn't have an attack in the queenside and instead white has a strong centre. As for the space "advantage" you say, Black has most of his pieces at kingside so there is not way to capitulate the "advantage".