Agree, that would seem to be the problem. Have you checked to see if it spins fast between wash and rinse? That would help narrow it down between control coming from the timer or brains as opposed to something else.
That seems possible.
If the door locked light turns on during the regular cycle, I doubt very much that is the problem. It seems unlikely it would indicate locked during the rest of the cycles and only go open during the spin cycle.
Given that it doesn't spin, I doubt that is the problem. But I would have checked that first, as it's easy to do. IF it has a a drain filter, make sure it's clean. I'd reroute the drain somehow temporarily, even if it;s to a couple of 5 gal buckets, so you can keep an eye on it. IF this is one of the newer front loaders, they don't use much water. Also, seems unlikely it could be a drain issue because I would think if it couldn't drain, you'd have other symptoms, like soapy clothes.