No, you are not required to turn right at a red light where it is permitted. You have the option to either turn right after stopping and ensuring it is safe or to remain stopped and proceed when the light turns green. Always check for pedestrians and oncoming traffic before making the turn.
You can turn left at a red light only if signs specifically permit it. Usually, only right turns are permitted on red when clear.
Unless the lane is specifically marked as "Right turn permitted without stopping", yes. In any case you would still be required to yield.
You must still come to a complete stop. Even if there is a flashing red light you must stop.
Yield to the right-of-way vehicles, until clear then turn cautiously
Generally, a red light means "stop," but in many cities, it is permitted to make a right turn on red, with caution, if it is safe to do so (no other cars are coming). But as you have noticed, there are exceptions. If you see a sign that says "no right turn on red," that means that specific red light requires a full stop, and you may not make a right turn until the light changes to green. In other words, if you do not see any signs, you generally are allowed to make a right turn on red. If you do see a sign, you are not allowed to make a right turn on red.
Stopped!
For left-hand drivers, depending on location: usually at any signallized intersection unless it is NOT permitted by a posted sign or a signal light that controls right turns. For right-hand drivers: never.
They are not exempt from such restrictions if they exist. If right turns on a red light are not permitted at that intersection, then that is applicable as much to a school bus as it is any other non-emergency vehicle.
You must still come to a complete stop. Even if there is a flashing red light you must stop.
If there is a light labeled right turn signal and it is red, you can not legally turn.
The left turn has right away