It really depends on the surgical procedure they are doing. I am "guessing" the surgery is the Pelvic Osteotomy which is commonly done in young dogs.
Patients have a restricted leash exercise for about 6 weeks...and duration can be increased based on the patient's tolerance. They usually do one hip at a time...so the dog is immediately up after surgery and over the course of a few days starts toe touching with the surgical side.
If your concern is carrying the dog and urination/defecation...you will be amazed at how quickly this dog gets up and moves for you. What you have to remember is that this dog (by the actions you describe) is already in a fair amount of pain...so after a couple days...(and pain medications) you should see a good bit of improvement. The biggest problem is people spending the money and not doing the post surgical care and the dog messing up the surgery.
I just want to stress, if at all possible, get the surgery. I can not stress how horrible it is to see a 3 yr old dog that can no longer get up...but is otherwise healthy...be put to sleep or made to suffer a more difficult surgery, just because owners wanted to wait it out. It will not get better with time!
I am so sorry your family is having to deal with this...it is stressful, and during this phase of the dogs life you should be enjoying the dog not worrying.
My thoughts and prayers are with you as you proceed on this journey with your dog.
Dr.Vamvakias
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Dr. D.A. Vamvakias
Veterinarian