I'm planning on running som level runs in the future but I'm wondering if they are played in story or freeplay meode?

It would be in free play mode so you would beat the game first, then go back and select the mission that you want to run.

I think it might be easier for players to understand if the category was changed from NG+ to Freeplay - Also, would freeplay allow red bricks or not? I've always been on the stance that Freeplay should show skill and not use them rather than using invincibility to just run through everything, but it's something that I guess should be discussed.

Changing that is not an issue but there needs to be runs submitted before any further action is taken to be honest, I created the IL's and the people who said they would do runs never did. If anything I would just make with/without Red Bricks a value to select since there isn't much interest in this game for it to matter too much. Having run the game though, there is more potential for time saves and skips with invincibility so you would want to consider that as well, going for these time saves can take some practice to master.