It sounds to me like the parents have some issues? Address the Mother and if they are not onboard with discipline then terminate. I have a 3-year-old girl who is similar and I don't invade her space. She gets time outs for unacceptable behavior and if she refuses to say sorry she gets a time out. Period! When she cry’s because she is missing her Mom or Sister I leave her alone until she is ready to join the group. Not everyone likes hugs or likes to talk about there emotions. Introverted people need space!
It took my 3 year old a good 3 months to come out of her shell and now she knows her guidelines and boundaries.
If your not on the same page as the parents how can it work?