What Are the Chances that My Friends Dachshund Will Make a Full Recovery From Hind Leg Paralysis? ?

I agree with your friend's father, the dog needs to be euthanized. Your friend might want to get a second opinion from a canine orthopedic specialist, if that will make her feel better about the choice. Doesn't sound good at all. You don't give many details, like how old the dog is.