I think 14+ is right, you have a feral, or at least a semi feral kitten on your hands. It sounds so much like Rose and Rosebud during play and Rosebud when I picked her up. Rose taught her to be wary of humans :sad:.
How old do you think the kitten is?
Do you have a cat who is very good with kittens who may be able to socialize her?
When you pick her up, can you distract her with a yummy treat (if she is feral, raw chicken may be a good one) or a favourite toy so she is not totally focused on you, but will associate happy feelings with being picked up?
I can't see any vet cutting a cat tail except for medical reasons. Perhaps the tail was injured
.