The left turn has right away

You can turn right on a red light and a left red arrow. If there is a sign that says "no right turn on red" or it's a right red arrow, you can't turn right.

Typically, in states that allow right-on-red, it is legal unless there is a sign posted that says "No Turn On Red"

You can turn left at a red light only if signs specifically permit it. Usually, only right turns are permitted on red when clear.

Yes, they do. If you have a green signal to turn left, the opposing lane will have a red light. Even though vehicles may turn right on red (unless otherwise indicated), they are still required to stop and yield right of way on a red light.