Answering the second question first,
"2.Why, after 6th move black has better control over the center?"
Black's pawn on e5 is better than white's pawn on d3. Both sides have traded a central pawn. Black's e-pawn occupies a central square and controls another. White only controls e4. It's about the pawns. Pieces can move around and control various ...
First things first, this is an example of a very poorly played game, so Black gained its edge because of White's tactical mistakes, rather than by taking advantage of their extra space. This is a "fault" often made by chess authors when trying to illustrate a concept. They will explain the events in a game in a way that "fits their narrative&...
Black has more space. The e5-pawn gives Black four rows to move around in, while the d3=pawn gives White three (and the 4th rank is kind of a no-man's-land). That extra board room confers ease of the development: no problems for black bishops, while White's Be2 isn't so happy.
Later, the space advantage gives Black a chance to grow with ...f7-f5 plus ...e5-...
There're several long-term reasons to avoid 3.e4.
You permanently weaken the d4-square. This is dangerous because it's a central square. In the short term, Black can already play ...Nd4, plus ...Bc5 is another natural developing move that increases control of the square.
You stop your own long-term plans. You need some knowledge of how pawn structure works ...
After e3, white can follow up with d4 and will have a better center. The pawns on c4 and d4 control three of the four center squares while black only controls two.
e4 is similar to a Maroczy bind which isn't too bad despite the weak squares it creates. However, in this specific position, black hasn't committed to much so black has a lot of ways to combat it.
What everyone has said about the "hole" on d4 is logical; However looking at what Magnus Carlsen has done recently, e4 may be just as good!
I am referring to:
Usually e4 is played after Black has committed d7-d6 blocking the dark squared bishop as it was thought that Black is immediately equal after 4...Bc5 or 4......
With e2-e4, you create "holes" at d3 and d4 that can never again be attacked by pawns. Any enemy pieces that land on those holes have to be kicked out by your pieces, which is harder to do (because every piece runs away from a pawn job).
World champion Botvinnik used to set pawns on c4-d3-e4 which we call The Botvinnik Triangle. The world champion ...
You should pick one system that you like and matches the style of play suited to your personality. Then master that opening.
At your level there are dozens maybe hundreds of solid openings you could choose from that could work for you. Only a few would be best for you; but you need to decide that for yourself.
Damages white's pawn structure.
Paves the way for quick kingside castling.
Prevents Nc4, which would centralize the knight and attack e5.
Removes a potentially "bad bishop" for a knight that is about to become strong.
c3 pawn loses a defender.
a3 pawn is attackable.
Opens b-file for your ...
Black wrecks white's pawn structure
Black gains in development since he's trading an undeveloped piece for a developed one.
Although the center is sill fluid, black's dark square bishop is currently a "bad" bishop. in the short run, white's knight is the more active piece.
It removes the support of the c-pawn which opens up some tactical ...
The thing about loosening White's pawn structure, specifically, is that Black is enabled to develop ...Qd8-e7 or ...Qd8-d6 as a threat, when White's replies a3-a4 or Qd1-c1 don't help White.
Good chess is a matter of introducing unused force with threats.
While some books give a slightly higher numeric value for the bishop than the knight, an exchange of bishop vs. knight is not regarded as a sacrifice. It depends on the exact position if a bishop or a knight is stronger. So this is simply an exchange.
Bxa3 makes the white pawn structure worse. White gets an isolated double pawn, which is bad news for white. ...