You can shop around for lower rates elsewhere as you please. But keep in mind that the accident will still be on your record for all insurers to see and rate you on — namely, it will show up on your CLUE report, which all insurance companies use to access your claims history. On average, an at-fault accident increased rates by 42% in 2018.
Most people understand that if they were at-fault in a car crash then they will likely see an increase in their rates. Unfortunately, not-at-fault accidents can also affect the rate that you pay for car insurance.
Technically speaking, filing a claim would not result in a "point" on your friend's license because insurance companies do not issue "points" to the DMV. Points are the result of tickets and accidents where a police officer was called to the crash scene or wrote a citation.
Whether or not your insurance company paid anything for your accident, the incident itself will most likely be listed on your CLUE report. If a police officer wrote an accident report then it will likely be listed on your MVR as well.
Generally you can't remove a household member from your policy unless they have their own separate insurance coverage. You should have the option to exclude your son from the policy if you don't want him to impact your rate, but that means he will not be covered to drive any of the vehicles you currently insure.
The first thing I'd like to address is the concern about your daughter's portion of the premium increasing. Since you are both covered on the same policy the overall premium is going to increase if a claim is filed.
Our State of Insurance Report actually shows that the average premium in Wyoming of $1176 is below the nation average of $1323 per year. Since rates are partially based on the number of claims and damaging whether events in certain areas, it's possible that your rate is being impacted by things that are out of your control.
If your son hit a deer and that is what caused him to go into the ditch then the accident is likely to be considered a comprehensive claim rather than a collision. Comprehensive claims generally don't impact rates as severely, if at all, the way a collision claim would if he had hit another driver.
Unfortunately it is definitely possible that having a trampoline will cause your homeowners insurance rate to increase. It's even possible that having a trampoline could cause your insurance company to cancel or non-renew your policy altogether.