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.
Homeowner's insurance is to repair damage caused by an unforeseen event. In this case, it will most likely not be covered by insurance, especially because it is cosmetic damage caused by home improvements.
The cheapest type of car to insure is typically an early-2000s 4-door sedan. You want it to be a car that you are not making payments on that way you do not have to carry comprehensive and collision coverage.
When you file a claim for an injury, the party responsible for the injury would be the company to pay for the claim. If you had another injury from the second accident you will be able to file a claim.