I come to a four way signal light. The light is green. I'm signaling to make a right turn. The driver opposite me is signaling to turn left. Who has the right of way.

If I'm understanding correctly you are both trying to turn down the same block. If that is the case then as long as you were both waiting at the red light when it turned green and you hadn't just approached it then You would be the one to have the right of way.

Please do not double post questions. Your question was asked and answered here: https://www.askmehelpdesk.com/etiquette/driving-right-away-rules-605605.html

Yes, you would be the first to turn.

The right turn has the right of way, at a green traffic light ( not one with arrows and signaled turns)
Since you are turning within your own lane and the other driver will have to cross your lane of traffic to make the turn.