The first thing I would do is to try and see if you can get into a good dog training/behaviour class. If the vet is scared of her it is going to be very difficult to treat her for any illness she may have in her lifetime. I have had GSD's that did not like strangers (one was a rescue bitch) and she had to be muzzled everytime she went to the vets. This made life extremely difficult, especially when she had something wrong in her mouth or around her muzzle.
A good dog trainer will help you socialise her with dogs and people, don't think that she will love you any less if she gets on with other people and dogs!
As for spaying, if you are not going to breed from her (which I would not with her temperament) it is probably a good idea.
It will stop temperament changes around coming into season, like women dogs tend to get a bit hormonal and moody! My bitch had phantom pregnancies and used to nest build and would kill anyone who went near the 'nest' and she was normally a very nice bitch who would not dream of biting anyone!
As I say the first step for me would be a GOOD training class, as soon as you can, and both you and your dog will probably make some new friends.
Good Luck.