Does indeed sound like a typical rambunctious kitten. As cute as they are, they can also be persistently obnoxious
!! She should eventually grow out of it, but in the meantime do you have any wand or fishing rod style toys you can encourage her to play with? That will distract from your hands a bit, plus hopefully give her enough exercise to get some of her ya-yas out. The petting thing will come. It's probably still new to her, plus the fact that she sees your hands as toys confuses her.
As for introducing her to your older kitty, here are some links that might help the process:
Good luck! And we also demand kitten pics here if you have any.