Does white have a space advantage, or is the space advantage the same for both sides?
Is there anything black can do to increase his space advantage or to reduce whites space advantage?
The term "space" is typically defined as how much control one has over the position. In this case, we can see that White has a huge space advantage, with literally all of his pawns and pieces controlling the center. Black, on the other hand, has no control at all over the center.
"Control over the position" eeeh...
The term space is used two different ways:
1) Just counting the number of squares each side influences. This is the classical space when authors talked about space, force, and time.
2) Counting the empty squares behind your pawns that can be used to post pieces or maneuver through. This kind of space is quantifying maneuverability.
Space is overrated. The main reason why those positions are much better for white is not space, but rather control over the center. Having a cramped position is OK, though a cramped position is more difficult to play: there is no room for error. A position where the opponent controls the center however is lost unless you have a form of compensation.