Is there a way? if not i suggest such an option. i would like to sort out runs on my profile page by just hiding them instead of deleting them all together.

that would be a nice feature. but for now you could technically ask the mods of the game you dont want on your profile to edit your runs so that the username is starts with "[]" which wont show up on the name but the run will be on the boards and out of your profile

In multiplayer runs hosted by someone else, can the host do that themselves or do they need to ask the mods?

@yesseruser The person submitting the run can enter the partner’s name that way.