I'm just going to base this off of my speedrun that I just did (also dude where are you lol). I start the timer as soon as I hit solo, basing that off of the fact that when you land during Chapter 1, there's also a tutorial section so including both makes sense. As for when things end, I count that as when I press L to kill Hades, just like how with all the chapter splits I counted it as the end of that chapter when I landed the final blow on the boss.
I think the only thing that shouldn't be allowed is being able to speedrun a game with an already-started file, as that's currently allowed in the guidelines. Besides that the game lends itself quite nicely to the format.
Could somebody, instead of just going through solo immediately, could they go through together mode, get a good weapon to carry them through the game, then proceed to start the run? Or should we say that you cannot go into Together at all during the run? Because doing so sounds like it would be unfair.