But – Is this stuff playable?? (Part I)

Of course not.

The Jerome Gambit (1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5 4.Bxf7+) has many refutations.

I'm glad that's settled.

Maybe a more useful question would be --
Under what conditions might the Jerome Gambit be playable?

In casual or blitz games among "average" players , perhaps -- when Grandmaster Nigel Davies' words from his Gambiteer I (2007) are relevant:

Having examined literally thousands of club players’ games over the years, I have noticed several things:
1) The player with the more active pieces tends to win.
2) A pawn or even several pawns is rarely a decisive advantage.
3) Nobody knows much theory.
4) When faced with aggressive play, the usual reaction is to cower.

I like the sound of that.

Also, what do you make of the following position?

White, down two pieces without compensation, is lost, right?

Actually, the game is Morphy - T. Knight, New Orleans 1856, which ended in the first player announcing checkmate in 17 moves.

And this position?

Morphy - Maurian, New Orleans, 1858, a win for White in 15 moves..

Ok, you're catching on. It's possible for a stronger player to give a weaker player odds of a piece or more and still have a fighting chance.

Just like someone could give "Jerome Gambit odds" in the right situation..

One more position: White is lost, right??

This is a trick position.

Vazquez - Giraudy
Mexico, 1876
1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5 4.Bxf7+ Kxf7 5.Nxe5+ Nxe5 6.Qh5+ Ke6 7.Qf5+ Kd6 8.d4 Bxd4 9.c3 Bb6 10.f4 Qf6 11.fxe5+ Qxe5 12.Bf4 Qxf4 13.Qxf4+ Ke7 14.Rf1 Nh6 15.Qe5+ Kd8 16.Qxg7 Re8 17.Qg5+ Re7 18.Rf8 mate

Andres Clemente Vazquez, the Mexican Champion and editor of La Estrategia Mexicane gave both Knight's and Jerome Gambit odds -- and still managed a checkmate in under 20 moves!