Actually as long as you have water pressure you can. It depends on where your waste water goes. If it goes to a sewage station that is also without power things can get nasty quickly. I have lived in places that had a septic tank and it was fine as long as we had water pressure. If you have a well, then it may be difficult. If we knew a storm was coming we would fill our tub so we could use a bucket to make the toilet work. One storm our power was out so long we used water from the neighbors pool.
If you have water in the tank, you can flush it. If you have no water due to a storm or power outage, you can dump a bucket of water into the bowl and it will flush. That's why it is advisable to fill your bathtub and have containers of water if there is a possibility of a power outage. Then you can always flush the toilet.
Yes. As long as the tank is full and/or you have water line pressure, the toilet will flush. Keep in mind that if you lose water pressure, the toilet will not flush again. In some cases, if the bowl doesn't refill with water and seal the trap, sewer gas may back flow into your home and create a health or explosion hazard.