Even if they have their own car and policy, if they live in your house your rates are affected.
Not true. An insurer can't charge premium for someone that's already paying premium. If they drive your cars, it's best to leave everyone on the same policy. Plus you have multi car discounts and your years of driving experience working to counteract their inexperience, which does help some with the rates.
I have to ask, if he caused $5000 damage to your car, what did he hit? If he caused damage to another vehicle or property, do yourself a favor and don't hide this from the insurance company. You're leaving yourself with quite a liability. And as someone else suggested, don't underinsure yourself trying to save money.
Ask if there are any discounts the kids can get...student, driving courses, etc. They may have lost eligibility after their accidents, but it doesn't hurt to ask. And my totally unsolicited two cents, this may also be a really good learning opportunity for your young drivers who will someday be in charge of their own bills and insurance. Showing them the financial consequences of having an accident and even having them work to contribute toward the premium may go a long way the next time they get behind the wheel. Accidents are accidents, but I can tell you I drive a lot differently in a parking lot after having an accident I was partially responsible for.