First of all insurance companies don't cancel policies without notice. Second, it doesn't matter if you had current coverage or not if someone else was responsible for the damage to your car. File the claim with the insurance company that was covering the at fault driver.
Yes.
i do my errands on time
That is the correct spelling of the plural noun "errands" (tasks, chores).
I'll be running errands for the rest of the day.
Generally no, and it could get a little interesting if you are running errands for your employer. Your personal auto insurance policy does not cover commercial use. You need a commercial lines policy, not personal lines.
no
The expression is not idiomatic. It means exactly what it says. To be sent on ( or for) errands means to be out on a shopping trip, or such like, for someone. Mother sent me on errands to the grocery store and the dry cleaners.
If you "send someone for errands" you're giving them a meaningless task to do just to get them out of your way.
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Call the insurance company and have them add your name. I drove my mother around and used her car for errands for her and never worried about the insurance. I have my own coverage for my car and she had hers, so if anything had happened I was covered. Often your own private car insurance goes with you when you drive another car.
Of course they did.