I've just built my first store and I wanted to assign Codsworth to run it... I click on the store in workshop mode and it says this resource is now assigned but Codsworth doesn't move... Is this a bug?

Try making sure there is adequate room around the workshop for him to actually move there. Picking it up and placing it again. There's probably something a little too close by.

I haven't tried with a companion. I've had it happen with a settler, who were standing 10 meters away, tho, but the selling still worked.

I can't even assign my followers anymore..