OK, wow, if she was walking at 8 weeks, that is extremely good. You guys are really fortunate. I have always thought it is true that gaining strength and balance has something to do with the ability to toilet. If she is tensing up trying to stand or hold a position squatting, it must make it harder for her to relax and do her business. The other thing is, if her surgery was mid back, that tends to create tight sphincters, as opposed to a lower back injury which tends to cause more relaxed sphincters.
I don't know the answer on the water. It may be exactly as you say, it's just more obvious now. If she is or was on steroids, that can make the dog thirsty, but if she is off the medication, she should go back to her normal drinking habits.
It's true about the diaper, it won't hold in #2. Or rather, it shouldn't because she is a female dog and you do not want solid waste collecting inside a diaper where it could cause a urinary infection. Stool quality is important with a paralyzed or recovering dog. If she has soft stools, that could be a mess. I've had really good luck feeding Science Diet w/d dry kibble, as far as producing good stools for an incontinent dog, whether paralyzed or just senior. The stools are low odor and firm and not sticky. If my dog doodles in bed, I can pick it up with a kleenex, and she also stays clean.
There is one option for diapers, but it may be more fuss than you want to do. You could put a disposable diaper on her to collect urine, but cut a tail hole in it so the stools fall out. Over that you could put a diaper cover with a locking tail hole. That way if she doodles, the waste does stay inside the diaper cover and your bed is clean, but it is not contacting the urogenital area where she could get an infection. In the morning it will be easy enough to remove the diaper cover and throw away the disposable. This is another time when stool quality matters. If she has firm stools and drops one inside the diaper cover, it won't be a big mess. Here is an example of a diaper with a locking tail hole.https://www.samsdoghut.com/ecommerce/ul ... wraps.html
I don't really know of any exercises, but time is on your side. Recovery from paralysis continues not for months but for years, and more than one person here has noticed that sometimes just the tiniest improvement can be all it takes for the dog to be able to do something he could not do before. All of her daily exercise and all of the general improvement in her state of health from good nutrition, etc., will help. If you live in a climate where the weather is getting warm, you might try taking her swimming when you have time. That helps with overall conditioning.