I use to have a cat with no tail too. People up the street from my mothers have 2 Gray Hounds, Each one is missing a leg. Its amazing how well they get around.
I was given the option of taking off Maxie's leg, but I couldn't do it. I know she's in pain now and again but it's still useful to her.
Neither of my mutts are desirable by most standards, but they're my mutts and they at least pretend to like me, lol. I wouldn't trade them for anything.
Here's a pic of Merlin in the shelter.... Undernourished, dirty, bowed legs, completley untrained, had to have a rope surgically removed from his neck (debrided), stomach was full of garbage AND he had kennel cough... but, I saw greatness in him. :)
BTW, Maxi was no prize either, lol. She looked great but her owners couldn't handle the attitude....