Hmm... I haven't heard of this Italian Gambit with 4.d4(?!) before. I think the idea is similar to the 5.d4 gambit, but here after 4...Nxd4 5.Ne5 Qe7(!) like you said, I can't find any great line for white. In all cases black is a solid pawn up/left with a positional advantage. I haven't checked these with the engine, so improvements might be possible (especially around 8...Qe1+(!)).
Can Italian gambit be accepted with Nxd4 ?
Sort:
Stockfish is correct. After 4...Nxd4 5.Nxe5 Qe7, Black gets a small advantage after 6.Bxf7+ Kf8 7.Bxg8 Kxg8. And a larger edge after 6.Nxf7 Qxe4+ 7.Kf1 Qxc2
Interesting
This approach for White has a bit more punch after 4.O-O Nf6 5.d4
Laskersnephew wrote:
This approach for White has a bit more punch after 4.O-O Nf6 5.d4
True. Now 5...Nxd4? 6.Nxe5 is good for white.
But there is also 4.0-0 d6, and now there is no Scotch/Italian gambit in sight.
Casually, 4..d4 Nxd4 5.Nxe5 Qe7 is "good for Black", if one believes an engine's fast output.
However, I think that white is not worse in this really murky position.
So, it is rather a matter of taste going for that small mess as white, or not.
Regarding #2: It's still a mess after 6.Bxf7+ Kg8 7.Bxg8 Qxe5 but I'd rather be white after 8.Bd5 c6 9.c3! cxd5 10.cxd4 Qxe4+ 11.Be3 Qxg2 12.Kd2 Bb4+ 13.Nc3.
Wikipedia https://en.wikipedia.org/wiki/Italian_Gambit says that "Nxd4 is inferior, as it allows for 5.Nxe5, attacking f7 pawn".
https://www.chess.com/explorer?moveList=e4+e5+Nf3+Nc6+Bc4+Bc5+d4+Nxd4&ply=8&origMoves=e4+e5+Nf3+Nc6+Bc4+Bc5+d4+Nxd4
However, Stockfish seems to like the position, as Black can counter 5.Nxe5 with 5...Qe7.
That is why I think the analysis on this move should not be discarded.