Adam, as well as every retro character, is available from the beginning of the run in Story Mode once they've been unlocked. I'm not going to sit and argue back and forth about why Story Mode should exist in the leaderboards. It should, and it should be self-evident that it's built into the game as a division in the game and that Story Mode gives players the ability to take a strategic death to save time. What really is evident to me though is that Story Mode isn't going to become a category because the community is afraid of cheaters. Fair enough, it still SHOULD be a category because there are legitimate strategies that differ between Arcade and it. As it stands the only way that's going to happen though is if someone who's respected and highly skilled fights for it, probably by setting a time that undercuts the highest arcade times with it. There's no incentive for any of them to try, though, unless they can drop the time by something ridiculous like twenty minutes, which is probably not possible.

One thing I will note as an aside is that no, there definitely should not be a combined list for all characters. Each character plays so differently that if the leaderboard is not segregated by character, or even if there's a half-measure to put them together, there will be no incentive to play as certain characters and learn their strats because it will be nothing but Adam and Shiva runs, or more likely, just one of them. Comparing high level Floyd playthroughs to Shiva playthroughs are completely different things, and it's essentially unfair to compare them.

any game over should automatically invalidate a run. There's no situation in which it would be 'fast' to game over.

Is there any reason there's not a Story category? Considering the difference in how lives work between it and Arcade it feels like there would be some different strategies possible.