Sounds like a typical 3 month old to me. I have 2 puppies that are now almost 7 months old. The good news is they do slow down a little in a couple months, but mine can still get wild.
When she gets too rough yelp so she knows she is hurting you. Mine have each other and my older Yorkie to teach them to not be too rough, but if I yelp they would back right off and bite me more gently. My puppy (the other is my daughter's) has his mouth on me all the time and I rarely have to tell him any longer to be gently.
I would let them chew on my hands though as long as they did not hurt me. This is the age to teach them. The zooming around though is typical Yorkie puppy behavior. Enjoy it and get them to chase a ball or play tug of war with a toy. Take them for a walk around the block and wear them out. They will slow down in a month or two, but not too much. I had 2 at 10 weeks old and for about 3 weeks I wasn't sure I was going to make it! LOL. Now, things are amazing.
Good luck and enjoy your new baby!
__________________  It's raining Yorkies here! LOL Teek  ,Rowan  , Raksha (Grand Puppy)  , Raelyn  |