Yes, you could severely damage a transmission by shifting into reverse while moving forward. But the odds of the transmission actually going into reverse are not good because there are safety mechanisms in place to prevent it from happening.
You could break the tranny and would have to replace it.
Your arctic cat 580 will go in reverse and not forward because the reverse linkage is okay while the forward linkage is not.
Come to a full stop, put on the handbrake, change the gear to first (or drive) and begin driving forwards as normal. Do not try to put the car into a forward gear while you are still travelling backwards.
On the vast majority of modern cars today it is impossible to actually shift to reverse while driving at high speed. Modern transmissions have an override safety device that prevents the transmission from going into reverse at any speed forward that would harm the transmission. You could put the selector in reverse, but the transmission would not shift to reverse. On older transmissions without this feature, you would probably destroy the transmission and possible have an accident.
The rear of the car will always go in the direction that you turned the wheel if you are in reverse. It's just like driving forward. If you want to go forward and right, turn the wheel to the right. If you are in reverse and want the rear of the car to move to the right, then turn the wheel to the right. To answer your question directly, the rear of the car will go to the right.
park in reverse while going forward
Shift to reverse
Check the fluid for a burnt smell or odd color. While driving the car with low transmission fluid its possible that the bands were damaged.
Bang depending on the vehicle, you will ruin the transmission.
That is what is called an optical illusion.
Inspect your brake pads. I would suspect they are worn out.