For the 2 - 4 player new game runs, do the other players have to do new game as well? And do they need video proof as well for the run?

All players are required to be playing a new game/newly created character at the time of the run. And video proof of the others players is required from their perspectives.

It would be rather silly if no one else used new characters, and against the point of proofed runs if no one had verifiable video that they did the run properly, its also good practice and should be suggested that everyone have a recording of a run from their perspective.

If you have other questions or the likes you can join the Discord which is active in all forms of Monster Hunter Speedrunning.

https://discord.gg/z8suWUX