I've noticed that a lot of the runs in this category don't follow the rules saying that 'Timing starts once "Custom Night" is pressed'. The majority of runs start the timing once the custom night screen fully loads, including at least the top 5 runs... When I retimed the world record, I found the actual time is 800ms, not 240ms.

If the Twitch app doesn't work on mobile, you could always use https://twitch.tv/ and request desktop site

Please refer to this guide: https://www.speedrun.com/coloring_pixels/guide/lackx

Yes

I'm not sure about the load times, but since it says that 100% No Skips start when "New Game" is pressed, I would say that you cannot start on any night other than Night 1. This is probably also because setting all of the animatronics AI level to 20 is the main part of the run that differentiates your time to someone else's.