Are we allowed to use badges for speedrun?

I'd say no. Not everyone will have access to them (personally, I've only got two despite playing this game nigh-constantly before I picked it up for speedrunning, just because of the way I tend to play survival mode), and a couple would make a pretty big difference for speedrun purposes.

Can you even enable them in Story Mode anyway? I'd check but work.

Pretty Sure you can only allow badges for use in survival mode so this question is null lol.

Which is amusing, because the only badges I have, I have because I started running this game, and grinding out the movement and the outdoors-time in story mode pushed me over the top for those badges!

I think badges are only for survival mode. But do not know if they have effect in challenges.

Yeah I noticed that we can't use badges in challenges. But, who knows, maybe one day there will be a category for survival mode.