How to reach an open position in the 4 Knights Game without exchanging too much material? I believe that 4 knights opening is the most popular opening, especially among average players. I play it a lot, that's why I want to analyse it. I'm an average player, so I hope you can help me.

Some people like to play 4 knights game like this

[FEN ""]
1. e4 e5 2. Nf3 Nc6 3. Nc3 Nf6 4. Bb5


But I don't like to exchange my bishop for a knight, early on, so I play something like

[FEN ""]
1.e4 e5 2.Nf3 Nc6 3.Nc3 Nf6 4.d3 d6 5.Be2 Be7 6.O-O O-O


This is just an example where both player are trying to control the center, develop their minor pieces and castle. So you can think of any position, since most of the 4 knights game positions will most probably look a lot like that.

Both sides castled, now, how to continue without exchanging too much material? I'm just an average player, what I would do is 

[FEN ""]
1.e4 e5 2.Nf3 Nc6 3.Nc3 Nf6 4.d3 d6 5.Be2 Be7 6.O-O O-O 7.d4 exd4 8.Nxd4 Nxd4 9.Qxd4 


That's what I would do but I don't like this position, black is going to chase my queen, which happened to me a lot of times, and I'll end up moving the queen, losing space tempo, space and advantage of being one move ahead. If black played c5 I'll have to bring back my queen, if Qc4 then Be6. Materials are equal, black gaining tempo while developing his pieces. And both sides lost too much material to open the position.

I don't see anyway to continue the game if white didn't play d4. And in many games you'll end up swapping queens too early in the game if the d file was open.

So is it possible to play the 4 knights opening without trading too much material in order to open the position? (I'm asking about black and white.)
There isn't really anything wrong with what you show as an example in my opinion. White is in the superior position. Just because the queen is attacked once does not mean that you are going to lose a tempo. If Black moves c5 to attack your queen on d4 then that is a blunder in my opinion, and can be countered with qd3. In this position stockfish gives White a +0.64 advantage with 23 moves of depth. So I would say that you are on the right track. Continue to dominate that center, and then look towards exploiting Black's weak pawn on d6 by getting your f rook involved and developing your other bishop. Game on!