I think that a lot of the neediness is adjustment...she's been through a LOT in the last few weeks, and she may be a little overwhelmed by too much new territory all at once. Can you try giving her a day or two in one room with her food, littlerbox, toys, and scratching posts, going in often to visit, play, and cuddle? Then gradually reintroduce her to the rest of the house so it's not too much all at once, and see if she does better once she's had time to figure out where everything is. From shelter to spay to sickbed to giant new home is a lot to take in all at once.
Leave her a radio on low volume when she's alone (my cat likes NPR).
Buy some cheap carpet runners or welcome mats for scratching. Some cats prefer to scratch horizontal surfaces, so if she does, just provide her with what she likes, and when she scratches other places, gently move her to the "right" place (you may have to scratch it yourself to give her the idea). Find out the patterns to her scratching and what she likes, and provide her with more attractive alternatives. Also, you can try a product called "Soft paws," little rubber caps that glue onto the claws so that she can still scratch, but not damage stuff.
Read the article on Play therapy at LittleBigCat, you may just not be engaging her in a way that works, and she sounds like she could use the play (ankle attacks are a dead giveaway).
The biting issue...some cats are easily overstimulated by patting, and it's very rare that a cat seriously bites without some kind of warning, so you just need to learn the warning signs (tail, ear, and whisker twitches, skin twitches, and direct staring. Try only patting for three strokes (or two, or one, whatever works), no matter how much she wants attention...this will keep her from getting overstimulated, and if you do the play therapy, you may find that play aggression and overstimulation have been dual problems feeding off each other. Don't yank your hand away or yell when she bites (providing it's safe), as if she needs hunting-type play, that will just get her more worked up. Just go limp, remove your hand gently (maybe saying "no bites") and ignore her for a few minutes. I think this is a secondary issue that well-applied play therapy will help with quite a bit.