You'll save on fuel bills if you don't, though, if your water heater is a gas heater without an electronic pilot. The pilot alone will keep your water quite hot while you're away on a vacation or at home by yourself.
If it's electric, it's good enough to turn the controller down to its lowest temperature point and leave it on until you're ready to use fully hot water, and it will perform the same function as Walt points out, for what he says is true about a constant temperature.
However a deviation by a few degrees will work too, which is why pilot-only operation works, as will lowest-point temp setting.
Yes. For regular use water heaters are meant to stay on. Consider how much energy it takes to re-heat all that water in the tank, and the inconvenience having to wait for hot water. So, yes, final answer.