This accident would actually be paid out under your collision coverage because you collided with the tire. Because you only damaged your own vehicle in this single-vehicle collision, it would also be considered an at-fault collision claim.
Every car insurance company is different and has different guidelines so I can't speak to what your companies policy on refunds are specifically. Typically if you cancel your policy within days of starting it, you would receive some form of a refund.
In this situation, you do not need to get the insurance first before you buy the car - you will do them at the same time. I would find a vehicle you are interested in and start shopping for quotes on that vehicle so you can get an idea of how much it will cost.
An at-fault accident in Ohio raised rates an average of 48% in 2017 according to our State of Insurance. Now, that increase will stay on your insurance record for 3 years - meaning, you will continue to be charged for 3 years.
This type of claim would actually be considered a collision claim. Although to answer your original question, a comprehensive claim wouldn't have as much of an effect on your premium as a collision claim would.