Reeb, some kitties don't calm down until months or even a year after their spaying/neutering!
Gerona, my white cat, calmed down about a month after she was spayed. But Galina was spayed in October and she STILL hasn't calmed down.
The running around and pouncing at imaginary things sounds like normal kitty behaviour to me.
Our cats have spurts of energy where they will TEAR through the house, leaping off the back of the couches and sides of the walls and just zipping around like balls in a pinball machine! My husband and I call this "kitty war games."
As for the biting, the link Candycane gave is very good.
You just have to learn to understand kitty body language.
Hunh ... I would NEVER put a cat on its back to try and get it to stop being aggressive. This works for some dogs
, but dog behaviour is much different from a cat's (as we all know
). I guess the person who suggested that figured, "Well, if it works for Fido, it might work for Puss, too."
We hold Gerona on her back like a little baby and she doesn't mind, but if we ever tried that with Galina, she'd eat us alive.