I have had this system played against me more often than main-line two knights. Is it considered a respectable opening or not? Is there a refutation to it? I have a fairly high success rate against it.

I am not sure if there is a "refutation" but it gives White a free tempo in the opening which means that he has an advantage.  I would continue like an Italian Game with 0-0, d3,Nc3 etc where ...h6 is just a wasted move. 4.Ng5 is sharp but not a winning line. 

This is one of my games against this system (obviously I am not saying this game constitutes "proof" of it's unsoundness)1.e4e5

Bc5, with the queen eying g5 is a much more respectable way to play.  Then Nf6 can be played, and if Ng5, 0-0 and white can't do anything.

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if Bc5, iw ould go with gambit on b4, then continue with c3 and then d4. blacks usual respone would be capturing the gambit, then moving the bishop back to a5.

What about 4.d4 ?

That's something that wouldn't work if black hadn't thrown away that tempo.

yes dilling3r, the evans gambit exists and is very very scary.  I score horribly against it.  But still, h6 is simply a timid waste of a move

it's a quite good move, though black still has another very good alternative in the Fried Liver (the name of the 2knights from black)

Strictly speaking, this is not "unsound", because there are games in the database. But it is really close to being unsound, as black uselessly goes another tempo down. 4.d3 line gives black equality.

The strongest response to it is 4.0-0 ... 5.d4, or 4.d4

a better anti-Ng5 opening is the hungarian defence.

I would probably rather play h6 as posted in the beginning than play the Englund Gambit (1.d4 e5?!)

Thats true, I forgot all about the hungarian.  Prehaps I should play that instead of getting my butt kicked by the evans gambit every week...

Also, dsarker

"Strictly speaking, this is not "unsound", because there are games in the database"

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 This move order that you put, I have never seen...normally, after 1. e4 e5 2. Nf3 Nc6 3. Bc4 Nf6 4. Ng5 d5 I have never seen anyone immediately play 5. Nxf7.  They normally play 5. exd first.  I don't think I would want to play 5. Nxf7 honestly.

And BTW, 3...h6 isn't terrible.  It is adventurous.  Many people like to experiment with these kind of systems.  For one, g5, a critical square in this opening, is covered.  Two, Black is already preparing a kingside pawnstorm if necessary.  Three, he is getting rid of White's theoretical Fried Liver crown and venturing into new territory.  Four, even the Evans Gambit lines are not possible now - once again, new territory.  White needs to try something new. 

And the Hungarian is good too, it reminds you a bit of some Philidor or Ruy Lopez lines...

If you want crazy, learn the Wilkes-Barre!

Here's the one game I played here with that. http://www.chess.com/echess/game.html?id=6736688

I tend to counter that h6 move with something like 4.d4 exd4 5.0-0 and play it like a Scotch gambit with an extra tempo. Another alternative would be 5.c3 dxc3 6.Nxc3 likewise going into a sharper opening where the tempo loss with h6 hurts more than in the solid systems.

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 I think people who do not use a database often frown when I mention it.

Database I am consulting is collected from games of rated players. Some master actually played it and pulled it off. I was also told this by 66_Mustang (I also previously thought h6 was unsound) and he cited an actual rated game to prove his point (I do not remember the exact reference now).

But usually black players here do not know the correct line. and come to grief.