I played a game recently in which a very odd opening took place. I was just wondering if move 5 was a sound move.

yep looks like a good opening

I think Black has compensation for the temporary king discomfort. His development should be pretty fluid.

Black should be careful about his open king, but other than that he has the 2 bishops, a central majority and a lead in development (after 6. Nxe4 ... d5). if he can consolidate his king position then he should do well and have some good pressure later on with the open f-file and with his 2 bishops.

maybe 5. d3 opening the diagonal of whites Bishop on c1 is more better than Bxf7 immediately, black reply with Nxc3 6. bxc3 - Bc5  7. Bxf7.

According to the Opening Explorer, Black wins about 61% of time in the games with this position.

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try the hungarian defence...

i.e Be7 instead of Nf6.

i had a decent success. with it...

White loses a bishop in exchange for ruining black's ability to castle.  The stats seem to indicate it isn't worth it for white.

Well, 4.Nf3 scores very poorly for white, but I think the best plan for white after 4...Nxe4 would be

5.Nxe4 d5  6.Bd3 dxe4  7.Bxe4

Compensation for the bishop? What do you mean? White gets compensation for the bishop with 6.Nxe4. You are talking as if this is a sacrifice.

Fischer played this when he was 12, he did pretty darn well!  The opening is not sound at all though, black gets a good center and the bishop pair, and white really doesn't have an attack against the king.  Unless you are fischer.  Regardless I would evaluate the postion -/+.  Harsh aren't I?

This is a post worth reading .

This opening is better for black.  This only works for white if black is intimidated by it enough to blunder.

Well, I thought about that when I read your post, but it did not seem like you were talking about the bishop pair.

The bishop pair can be an advantage, and something to work toward, but not all important. The right to castle and having two rooks active on the back rank are also advantages. In most games, the bishop pair is lost eventually, and all you can usually hope for is to transform it into some other type of advantage. I'm sure we can all remember a game where we gave up the bishop pair for a lot less than what white gets here.

Besides, if white cared so much about the bishop pair, he would not have played the aggressive move 1.Bc4. He would have developed it to e2 or g2, or maybe even d3.

White is down a bishop and has totally upset your game plan it all depends on what white does next  if he allows black get balanced and into a rythmn he may loose but if white keeps the pressure on he has a good chance is next move should be to get rid of blacks knight obviously and after that I would advance my kingside pawns rapidly and castle on the queenside if at all.In this game expect the unexpected, but thats Chess anyway.Taking risks is WHAT ITS ABOUT SOMETIMES AND SOMETIMES YOU SIT TIGHT just like life depending on your form your enviroment and your opponents. 

Best for white is 5.0-0 if 5...Nxc3 6.dxc3 and white has better development

Black shouldn't be too uncomfortable in this position. His King can always move back, and White doesn't have enough pieces out to start a real attack yet.

But not enough for the lost center pawn.

This ...Nxe4 variation is why Bc4 in the Four Knights is bad, and 4.Nc3? in the Two Knight's Defence leads to this same position.

I have played this before.  It depends on the player, sometimes it can through white a unexpected surprise.  Though I do not recommend this with a high caliber player.