Well, in this case I believe white with 1.d4 is winning. The blacks king side become more difficult to defend. White enjoy more space were the c file becomes weak! I though so, perhaps I am wrong.
What if the starting position were centrally symmetric?

A lot of my classmates set up the board like that. I think white's first move initiative would be much more evident in a position like that. However, launching an attack against the black king would take much longer as the white queen is not on the same-coloured square as the black king.

A lot of my classmates set up the board like that. I think white's first move initiative would be much more evident in a position like that. However, launching an attack against the black king would take much longer as the white queen is not on the same-coloured square as the black king.
Exactly. In chess, ...f6 is usually a bad move for Black because it terribly weakens the kingside. Now, it won't.

And c6 isn't going to be a bad move here. Whereas f6 in the original position weakens your position.

Is castling allowed for black?
For the purposes of the discussion, let's assume that black can castle to b8 or f8 - everything is rotated 180 degrees from white's position.

The chess engine will give 22(+0.36)
So, white is ahead by +0.36, according to the engine? Then I guess the position isn't all that bad for black, since the engine most likely thought that black couldn't castle (because under normal chess rules, he can't). So if we allow black to castle, then white's advantage is even smaller than that.
Basically, how it is sometimes set up incorrectly by beginners:
How would the game be different? Would white's first move advantage be more or less pronounced?