Ahh, someone beat me to the punch of creating a thread on a "dodgy" gambit for once... congratulations, Alumbrado
A) Okay, first this 4.d3 line. I'm fairly sure that such a move cannot be critical for the Schliemann's survival, though white can maybe keep his usual small edge.
In Tseitlin's old book on the Schliemann, he gives 6...Bc5 7.Qe2 d6 8.Qc4 Qe7 9.Nc3 Bd7 10.Nd5 Nxd5 11.exd5 Nd4 12.Bxd7+ Qxd7 13.Nxd4 Bxd4 =. Indeed, I can see no reason why this shouldn't be okay for black, so I'm not sure what there's meant to be to fear in this line. If there is some improvement (10.Bg5 a6 doesn't look like one), then black can still always fall back on 6...d6 which cuts out this plan, which again looks fine to me. Or otherwise, the plan of a reversed KGD with 4...Nf6 should also be playable.
Conclusion: White needs some improvement here for 4.d3 to be worrying.
B) 4.d4!? is, to my knowledge, unrefuted - apart from the critical 4.Nc3! this seems like the only plausible move for white to punish black's impudence with 3...f5. I'll only consider the main line, viz 4...fxe4 5.Nxe5 Nxe5 6.dxe5 c6 7.Nc3 cxb5 8.Nxe4 d5 (I've toyed with ideas like 8...Qb6 in blitz games, as no books to my knowledge offer any alternatives to 8...d5 and probably rightly so, but anyone playing this line as white will suddenly find themselves out of the forcing theory and, let's face it, a piece down. However, I doubt the move is any good in reality) 9.exd6 Nf6.
Now white, in reality, only has two moves - 10.Qd4 and 10.Bg5! - 10.O-O has been known to be weaker for a long time, as black can consolidate his material advantage.
A1) 10.Qd4 Nxe4 11.Qxe4+ Kf7 and now 12.Qd5+ Be6 13.Qxb7+ Kg8 14.Bf4 Rb8 (14...Rc8 15.Rd1!? Qd7 16.Qe4 Rc4 17.Qe3 h6 18.O-O g5?! and white won in Bigot - Hermann, 1990. 18...Rxc2 may be more critical, but white seems fine after 19.Be5 or 19.Rfe1.) 15.Qe4 Qd7 16.O-O is given as equal by Tseitlin, however I would think in practice white would have more fun from here, centralising his two rooks.
However, black can play 13...Kg6 as well. Now Tseitlin gives 14.h4 h5! 15.Bg5 Qa5+ 16.Bd2 Qd8 17.Bg5=. However, I find all this a little puzzling, since instead 17.Bc3!? looks like a move if white wants to play on, defending the d-pawn indirectly due to the threat on g7. Maybe the complications after 17...Rb8 18.Qe4+ Bf5 19.Qe3 Bxc2 20.O-O favour black, but my impression is that white should have play for a draw, for example 20...Bxd6 21.Rad1!! Bh2+ 22.Kxh2 Qxh4+ 23.Kg1 Bxd1 24.Qd3+ Kf7 25.Qf5+. The position is rich in possibilities, so I'll leave everyone work it out for themselves.
To my knowledge, 17.Bc3 has never been mentioned or played, so it may well be worth a shot since it's unlikely black will have a clue what's happening, and it wont be easy to find the best lines OTB. (If anyone comes up with a bust to this idea, please let me know!)
A2) 10.Bg5! (Tseitlin's exclamation, not mine, I prefer ?!) Qa5+ 11.Nc3! b4 12.Bxf6 gxf6 13.Nd5 b3+ and black gets an advantage - I find it difficult to argue with this this assessment.
Conclusion: White is struggling in the lines after 10.Bg5 if black is aware of the theory, however there looks like plenty of scope for experimentation after 10.Qd4 Nxe4 11.Qxe4+, and a draw if white wants it. Maybe there's improvements waiting to be found in the 10.O-O lines too!
I'll continue with C) 4.Nc3 in my next post!