As for someone to your left going straight thru the intersection... you, turning right, must yield to those going straight. However, there is not always traffic coming straight through the light. If someone on the perpendicular street turns right instead of going straight thru, you may turn right from your lane into their lane because there is a pause in traffic flow coming through the intersection.
That probably doesn't make much sense but the bottom line is that when everyone knows the rules and everyone observes them, right on red is mostly painless.
No they don't, They must come to a complete stop and yield to all traffic, including people turning left. At least in NJ, I don't know about other states.
so I guess you're right in the scenario that one direction has a green light with no green arrow and the opposite direction has a red light, but how often does that scenario exist?
If both directions have a green light, left-turners yield to right-turners because of traffic behind the right-turners and stop signs are always turn based.
Because someone who comes from your left may go forward, or someone who comes from your front mau turn left?
Where I live, we do not assume that we can turn right on red lights. Where and when that's possible, we have a blinking yellow light (basically meaning proceed with caution)..