Does it really exist? I personally think that it doesn't, it's just another thing like math which doesn't exist in reality. Is there evidence that time is an actual real thing?
Yes, we do. It has been measured since antiquity with watches and clocks.
Does "time" stop when the watches and clocks hands aren't moving? I don't think so. If I took the battery out of my wall clock, would "time" stop? No, but the clock would. Clocks do not prove the existence of time.