Is this Muzio-Gambit relative sound?

So far most lines Houdini 6 gives an evaluation of 0.00, which can be translated as unclear, and in an otb game between humans, I do believe Blacks Position, is incredible difficult to handle. 

Does anyone have some ideas/experiences if this Gambit can be considered sound?

This may be true for SF Evaluations, but not Houdini.

0.00 in Houdini Terms either means Repetition/draw OR unclear. if the line given shows no repetition it means unclear, because Houdini evaluations correlate with it's win expectations. So a 0.00 means it can't make any forecast, which side may win. 

AS far as your 'analysis' goes my question would be is what is the point of 5...g4?  It is not forced.

Black can simply play Bg7 & h6 in what order he likes and aim for a Hanstein or Phildor Gambit positions where Black has already played Nc3 which is supposedly better for Black.

5...g4 seems to be the most critical continuation as it forces the piece sacrifice.

Other continuations are not as dangerous for example:

3...g5 or 3...Nf6(!) are more testing than Fischer's "refutation".

Especially the latter is quite annoying to meet.

3...g5 is what a King's Gambit Player is Looking for. Now I abondoned the Muzio long ago in favor of 4.h4 followed by 5.Ne5, and it seems White has nice prospects against all Blacks defences. 

The Schallopp's Defence is recommended by Ntirlis in his 1.e4 e5 book. but after 

3...Nf6,4.e5 Nh5, 5.d4 d6, 6.Qe2 d5, he doesn't mention the move 7.Qf2 at all. After that move White can quickly conclude development and his pawncenter should guarantee him some nice attacking chances in the future.

I also find 2...exf4 3.Nf3 d5 and 2...d5 to be stronger than Fischer's supposed "refutation".  Not necessarily stronger than pfren's mentioned lines, but Fischer's line is a joke.

Because you are not Fischer...

Why Black should play the antipositional 6...d5?

Because Shaw says in his book this is best?

Ntirlis (and before him, Lokander) trust Shaw and push 6...d5, while in reality 6...dxe5 is both thematic, and strong. A summary of my analysis, a couple of years ago. A computer may draw the final position as white, but OTB things are quite unpleasant.

I think that while this line may well be better than Shaw's 5.Be2 (which simply does not work), it's nothing to be cheerful.

The positionally consistent line is the old move of Keres (3...Nf6 4.e5 Nh5) 5.Qe2!, simply because it is the only move that stops ....d6.

Keres lost his game against Alekhine, and the engines are discouraging/claiming  a large Black advantage, but I do think that white's position is quite playable, plus very much "king's gambit" style.