Your insurance won't cover you if get in accident and they will cancel your policy. You need commercial insurance even if you do one delivery. Call for a quote.
Your insurance company is giving you only half of the answer.
If Skip The Dishes is your primary income, then you will need additional coverage for business-use of vehicle.
If you already have a job and Skip is not your primary income, then your regular car insurance will cover you regardless of whether or not you are doing deliveries.
I have business use insurance that covers this. It does not cost much more than regular insurance.
I would do some research on different plans. Couriers and postal delivery people get insurance somehow.
No. You are considered a sole contractor. Skip does everything in its power to not pay for anything car related.