First off, welcome to the forum.
You state your kitten is only 2 months old
, too young to be away from his momma and siblings, who would be teaching him how to play socially. Anyways, if you want this behaviour to stop, you must stop all play with hands and feet. No exceptions and 100% consistency would be required for this to work
. You can redirect play with one of his toys. He will get the message. If he played too rough with his siblings, they would run away from him and not play with him.
You could also get him a playmate this same age to keep him occupied
A kitten is very active and needs stimulation to exhaust him.
As for the nails, please take your kitten to your vet to have a peek at, ingrown nails are very painful