Do I have to be live while doing a speedrun? Or, can I play offline and submit a run??

You can play offline. I recommend that u join the discord for updates on ruled such as these. It's located in one of the tabs to the side

If you submit a top three run, it has to have been ran live as a preventive measure against cheating.

BUt the problem with that is that people that are good enough to get WR but don't have the equipment to stream wont be able to get any where close to the top three.

"If you submit a top three run, it has to have been ran live as a preventive measure against cheating."

No, this rule has been removed again. They are now checking if you're run is legit, by just watching the run, asking you for videos of other tries, asking you for the world folder and more. After that, they decide if they trust you and verify your run or not.