Just on this point about ascribing things to a character. As it has been put on our screens that Lorca has been given the reputation and did not deny it, do we go by believing he killed his crew and saved himself?
I see shades of Burnham here. She also accepted a terrible reputation and the blame (both internally and externally) for some things she did and some things she didn't necessarily do.
I think it rings absolutely true that he blew up his ship. There's nothing to show yet, though, how and why he survived that explosion. It could easily be a case of being prevented from going down with his ship (for any number of reasons) but still blaming himself. Or, it could be that he scrambled to escape, knocking all other crewman out of the way.
We just don't know yet. I can assume, however, which you would prefer to assume.
The interesting thing, of course is who first introduces this character info in the show: Harry Mudd. For self-serving reasons, sow create division with Tyler. It was interesting that Lorca owned it and even said flat out that he blew the ship up.
Obviously there is more to the story and hopefully we will find out what it is. The fact that he essentially tortures himself everyday with eyeball injections when he doesn't have to, is a big tell that there's more going on psychologically.