This is the well-known Cochrane Gambit, named after John Cochrane (1798-1878), a Scottish player, playing in a romantic, attacking style, sacrifying pieces (around 1820).

The Cochrane Gambit (from around 1840) in the Petrov Defence aims at a strong center, easier to defend.    Bronstein liked it !

This is what we call the Carlos gambit.  Me and my best friend studied the Schliemann a little bit and he got this idea and played it against me when I played an Italian game with him.  Whether it has a real name I am uncertain, but it has been played before.

I think it's the Rousseau gambit but certainly Carlos gambit would be a fine name .

I feel like it is a pretty sound gambit.  We haven't studied it too extensively, but even with white playing well it seems to give black a playable game.

Yes, post nr 1 is the Rousseau Gambit, named after Eugene Rousseau (1810-1870) a French player who lived for a time in New Orleans, USA, --  and around 1841-1845 one of the strongest players in the USA ( so, before Morphy ).

This Gambit is "dubious".       There is an alternative name, Ponziani Counter-Gambit.   

For other readers of this forum : the Schliemann Defence is as follows : e4 e5 / Nf3 Nc6 / Bb5 f5 ( mostly followed by Nc3 -Berger Variation- Nf6 / exf5 e4 / Nh4 -Lasker Variation- ).