If someone pushes me or punches me am I allowed to punch them back by law?

Yes,

it's called self defense. Just make sure you hit them twice as hard as they hit you.
In theory, yes. However, it doesn't always work out in your favor. When I be 20, I tried to walk away from a fight. The girl kicked me and I walked away. Then she get in front of me and punched me. After that, I ended up being the one who get arrested because after I finally threw a punch, she didn't get another one in and she looked a lot worse for the wear. No witnesses, only her word against mine. They took hers. I got arrested, went to jail, have to be bonded out and had to go to court. I pleaded not guilty and had to step back. I explained everything to the judge and he still charged me. I had to rate a fine and court costs and got a suspended sentence. That did it for me. I realized that even self-defense fighting will draw from you in trouble. I've never put myself in that position again. Never will. Four hours in send to prison was four too many for me. yo i am not a lawyer, or responsible.

um, are you beneath 18? then yes. over 18 and you get into a fight, generate sure you get outta there before the cops show up!

peace.
Some states allow you to counter with equal force, others only allow self defense if you can not retreat.

In all states, self defense is an 'affirmative defense'. This manner it is up to you to prove you had the right to use it. This could mean serious legal issues, even if you are within the right.
More than likely yes, but you better be able to prove it was self defense. For instance, if the simply people around were friends of his they might say that you hit him first. Your one and only allowed to respond with just enough force to uphold yourself.
its call self defense

you can probably snap their neck and get away with it if the courts can prove you did it end in you thought your life was in menace
respectively state is different but most likely yes