I agree with tabs about the stress thing on your part... The only way you can make sure your baby is happy is providing all their needs.
Hitting and pushing is not acceptable, no matter what age and it needs to be addressed, which is hard with a baby so young, but she is old enough to understand no and understand the tone of your voice.
I think lack of sleep is getting to you both... I know you dont like hearing or thinking about this, but perhaps it is time to think about her own bed. In the beginning babies dont need routines, but once they get old enough routines etc help set boundaries and rules, it also helps them feel safe.
You are such a great caring person, there is no doubt about it, youve been through alot lately for sure.
The only thing you can do with teething is give a bit of gripe water and tylenol, and give them those frozen teething rings etc... im sure you do this already.