In most openings, pawns typically end up at some of the centre squares. There is really no other way to open up the long diagonals(c1-h6,a6-f1) for the bishops. However, how desirable is this? Hypothetically speaking, assuming there were some special rules in chess which allowed bishops to be developed along these long diagonals without having to push away the e and d pawns, would these pawns still end up getting pushed onto the centre squares?
Yes. The main reason to move your d- and e-pawns forward is to control the center with them. The fact that they allow the bishops to be developed is secondary. (Often bishops are fianchettoed anyway.)
The hypermoderns thought that pawn centers were weaknesses subject to attack. Thus came about defenses designed to "entice" center building while developing a way to undermine them. See defenses like the Alekhine or the King's Indian where this is exactly what is done. This is still an undecided question, although current theory believes that occupying the center with pawns is a strength, not a weakness.
Pawns that stand on or guard a central square (e4, d4, d5 or e5) are said to control the center. For example, the white d-pawn has the its maximum center control on d4. Yet, after d2-d4, it loses the potential control over c4 and e4 (the move d2-d3 is no longer possible). So yes, the d- and e-pawn will move forward. Yet, this can create weak squares in the center.