Yes, there are insurance companies that will insure you if you drive for a rideshare company. You will need to have a special endorsement on the policy that allows you to drive for rideshare and you are usually required to have higher limits.
In order to have a ridesharing policy with a standard car insurance company (such as Geico or Progressive), you have to have a personal policy already established. A personal policy is what the average driver in the US currently has.
If you have a police report with the at-fault party information on it, it's very likely Zipcar will contact them for the damage. However, I would check the insurance information you signed up for to see how you could be charged.
If you do not have a rideshare endorsement on your policy, they might deny the claim. Insurance companies like to know all the risks they could potentially be exposed to and ridesharing presents a big one.
Renting and borrowing a vehicle sound similar, but the way the insurance coverage works for both situations is different. Let's take a look: Renting - If you have an auto policy that includes comprehensive and collision coverage, that policy will normally extend to cover a rental vehicle as well.
Roadside assistance varies by insurance company. From my understanding, most towing and roadside assistance coverage offered as part of an auto policy either has a set dollar amount to cover a tow, covers towing up to 15 miles from your breakdown, or covers towing to the nearest repair facility.