| Has she always been this way? If not, there may be some pain involved here making her not want to be touched. If she's always been this way, no idea. Bad things happening early in their lives can affect them, but this seems extreme for that. And she's had human contact at least since she was 5 wks old.
As for the losing fur, that could be allergies. Or maybe just dry skin. How bad is it and how long has it been going on? Like just losing a tiny bit here or there, or losing bigger patches? If it's something like dry skin, giving her omega 3's can help. (Something like cod or salmon oil. They make kitty versions and you can go by the dose on it, but probably 1/4 tsp.) And I had a cat who was getting some scabs and losing just little bits of fur, and I put Vitamin E on those to help heal them, and also soften them up so the scabs themselves would stop itching. If it's allergies, you may have to do some food testing by eliminating all grains, seafood, beef. (There's a fairly long post on this I did under I believe "cat throwing up.") If steroids are recommended, try to do anything and everything else first, as those are bad news.
As for getting her to the vet, what I used to do with the she-witch I had was drop a towel horizontally over her, trying to aim to get the top edge just about at the edge of her on the head side. Then quickly grab in at the sides of the towel and wrap her. I'd have to be sure to catch all her paws in the towel. It was a knack I developed cause it was the only way to get her to the vet's safely. | 
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