As long as your license wasn't suspended as a result — or other violations that affect your driving record — you do not have to report fix-it tickets to your insurance company. Insurers typically will only rate against moving violations that show up on your motor vehicle record (MVR).
If your husband is able to get a cheaper quote due to his cleaner driving history, it would be a good decision to have separate policies. I would suggest you shop around for quotes as well to see if you can get a lower rate.
The only likely scenario in which a dropped ticket would still affect you would be if there was an accident involved. If the ticket was dropped, but your insurance company paid out to any party for a claim, the accident would still show.
I think it's worth trying to resolve it because a better driving record can help your rate quite a bit. If multiple insurance companies are showing that you have these tickets, then I would call the state and see what they have on your license — this could simply be an error.
Unfortunately, you likely won't be able to avoid any penalty for not having insurance. I would still recommend going to court with insurance — your judgment might be less severe if you resolve your uninsured status versus still being uninsured by the time your court date comes around.
Most companies do not consider non-moving violations as a factor when calculating your rate. But every company is different so you may run across one that will rate on non-moving violations, but it is really uncommon.
No, you don't have to report to your insurance company that you received a citation. But when it comes time to renew the policy, your insurance company will check your MVR (motor vehicle report) and probably raise your premium accordingly upon seeing the ticket.