exchanging queens first 3 moves?

Exchanging queens early is considered boring and lame. If opponent plays this against you it's better to play Nf3:

If they don't capture:

This is good for white.

If I were white I would exchange, because:

1) black would lose their castling rights

2) I’d gain a tempo, because the black King has to take the Queen and thus I would get a ”free” developing move

The disadvantage of losing castling rights isn't as significant when the queens are off the board as it's less dangerous. Also in the endgame the king will be better placed in the centre of the board.

Funnily enough, black is doing totally fine here. It's not even a disadvantage.

White did nothing wrong, it was black who resigned in an equal position for no reason.

True, losing castling rights is not as significant when the Queens come off the board, but it still matters. And also true that the King is better closer to the center in the endgame, but we are hardly in the endgame stage here as all the other pieces are still on the board and on their starting squares. The game is still in the opening stage.

And yes black can equalize in some lines, but does that mean it always happens? Sure doesn't. Does white automatically win? Of course not.

Right, but white maintains a much more significant advantage by keeping queens on the board.

A lot of bad advice in the comments, 2...e5 is an awful move and trading queens is very good for White. Stockfish rates the position +0.84 at depth 22 after 5. Bc4 Ke8 6. Be3 Nf6 7. Nd2. This is a bigger advantage than White has to start the game. Castling rights are still significant because all other pieces are on the board, and Black will have probably have to waste at least one more move to activate the h8 rook.

Allowing a queen trade early in the game where you have to recapture with your king is almost always a bad move. A lot of the advice here from lower rated players is simply off-base. 

Can you tell me why Stockfish11 gives 4. Qxd8+ as the top engine move then?

EDIT: I will add this: Basically if white does not exchange, they throw away most of the advantage they could have gained by exchanging.

You just played the engine moves without looking at other options. Ke8 isn't good of course.

This is more or less equal. Slightly better for white as they would normally be but nothing special. Black is just going to play Kc7 and solidify without many problems.

Obviously when the pawns have been traded Qxd8+ is the best move. When I say it's better to keep queens on the board I mean not taking on e5.

No I did not. I first figured out 4. Qxd8+ is the best move for the reasons I gave in my first reply.

Then after you stated that "white maintains a much more significant advantage by keeping queens on the board" I checked with Stockfish. Stockfish agreed with me. White does not maintain a more significant advantage by keeping the Queens on the board, but practically throws almost all of their advantage away for no reason.

Yes black can still equalize after the Queens come off, but they will have to work for it. If the Queens stay on, it is much easier for black to equalize (it is almost equal already at that point) and if white does something stupid like 4. Nc3 then 4... Qxd1+ gives a slight advantage to black.

These queenless middlegame positions just aren't that bad for black. Losing castling rights in these positions just isn't that bad. There is a similar variations against d4-c4 which is known to be good for black:

Black is actually known to be better here. That's why taking on e5 isn't the main move.

This is what I mean by keeping the queens on the board:

If you want to look at stockfish, you will see that it prefers Nf3 to dxe5.

I can't imagine how anybody would start a game of chess hoping to get an ugly position like that as Black. You have a symmetrical pawn structure with less space, worse pieces, and no winning chances. There is no way to create a meaningful imbalance in the position. And in fact this is why literally nobody plays even 1...d6 at the top levels, because it's about the most timid move there is.