I’m new to speed running and I’m looking and some of the category extensions and none of the rules seem to mention the daycare. What are the rules for the daycare especially for the no centres/no marts, no catches, and Baton Pass categories?

If it's not explicitly stated in the rules then it's probably allowed.

It's 100% allowed, but there's a 99% chance it's slower than what's currently used.

The daycare should be fine for no catches and baton pass. For no centers no marts it would go against "You are not allowed to buy anything, except: 1 Fresh Water and 1 Safari Zone visit", since withdrawing a pokemon from the daycare acts as a Max Potion that you buy for 100. However, I think it's perfectly within the rule set to drop off but never retrieve a pokemon from the daycare--there is no reason to do this but I suppose you can :)