4. exf5:
Schliemann Sidelines


4. Bxc6


4. d4 (Part 2)
if white does not play 0-0 move 10 after the bishop sac. All good sidelines are covered here (I think)

This is pretty comprehensive coverage; good work!
Regarding 4. Bxc4 - in the 5... fxe4 lines, what about 6. Qh5+ ? E.g:
There are similar lines in the Latvian and Calebrese Gambits, where Black can give up the rook and achieve some compensation due to White's isolated Queen, but not when White achieves best play. Furthermore, in those lines, Black still has a pawn on the d-file, which can provide cover for his exposed king if necessary.
Also, I've never come across the 4.d4 sideline before - is 4...exd4 possible, in the style of a reversed Falkbeer Counter-Gambit? E.g.:

As for the first one, Iwill have to check and anylize, I will get back to you.
On the 2nd one yes I think is it possible but in my opinion it is weaker, and dosn't gowith the goal of the schliemann.
I checked with a computer (lichess) and white goes from .3 to .7 after exd4, which is more than white haseven in the regular roy lopez.
So it seems safe for both players but white has an edge and fxe4 is more forcing and gets to the positions where we have preperation.
For the sake of not getting confused I will refer to the opening as the Schliemann.
The mainlines are 4.Nc3 and 4.d3 and I will get into those in another forum. But there are many other things White can try and it is good to know how to deal with them.