Most insurance companies will charge more for a turbo engine if the VIN identifies it as such — since it has more power and speed, the risk of a claim is going to increase. The same goes for luxury or modified vehicles.
There really isn't a global policy that can be used for what you are needing, nor does "temporary insurance" exist. The good news is most insurance plans are paid month-to-month and have very minimal penalties for canceling — if any penalty at all.
The rules for having a non-household member on your policy depends on the specific insurance companies rules and guidelines. Some companies do not like to insure a resident who does not like in the household.
The answer to your question will depend completely on the specifics of how the damage occurred. The gas station will pay for the damage if: If your vehicle was not moving and the brushes or moving parts of the car wash damaged your vehicle then the liability would be on the owner of the wash.
Most car insurance companies like to know everyone who is of driving age and has a license to be either listed on the policy or excluded. To them, it's simply peace of mind so they're aware of all the risks presented to them.