Car Insurance Costs
To answer your first question, will removing your son from the policy help decrease your premium, it depends. Some companies are going to charge you because the claim was made on your policy so your rates can still be impacted at renewal, even if you take your son off the policy.
For all their homeowners, Airbnb provides Host Protection coverage that protect the contents of the house from guest damages. You will also need to get regular homeowners insurance in addition to what Airbnb covers.
There are some steps you can take to try and get this fixed. Most insurance companies are using what is called a CLUE report.
There are too many unique rating factors that determine your car insurance rate for me to give you an exact answer. Based on your location, the average rate of car insurance in Colorado in 2017 was $1,435.
Coverage & Claims
Because most coverage options can be matched across insurance companies, I think what you are really looking for is the best customer service/claims support for the best cost. Right off the bat, I know Geico probably wouldn't be an option for you with a leased vehicle.
Dealerships and sometimes insurance companies thrown around the term full coverage but it really can mean a lot of things. Usually, it's referring to collision and comprehensive coverage.
Landlord policies can be actually be more expensive due to the risk of having tenants in the house. I would also make sure that any tenants you take on have their own renters plan that includes liability coverage.
If you have another policy already established with a car insurance company, they might offer a grace period. Meaning, a few weeks from when you buy it until you need to add it to your policy.
Coverage & Claims
I would assess the damages and if you can pay out of pocket, do so. While the claims department might disagree, this sounds like it would be an at-fault collision claim.
Most "standard" insurance companies don't advertise based on no down payments or being able to pay as you go. This is because they believe it will attract risky clients.