You are correct. Some injuries may take a while to become apparent. If you are still within the statute of limitations — and the injury was a direct result of the accident — you can reopen the claim to receive payment.
While I don't think your insurance agent is doing anything illegal, it's good practice for agencies to have fee disclosures. They can charge fees for services to make a few extra dollars but they should disclose it.
If you will be registering and living in South Dakota, then you will want to get an insurance policy for the address where you'll be living in that state. Insurance needs to be in the state that the vehicle is registered in.
From my understanding, if the initial check did not cover your travel expenses but was for the repairs then you are still owed for that and should reopen the claim with that company. They, of course, would need the info and receipts to show what expenses you had.
Claims departments should be marking the correct driver for a claim but other drivers pop up on reports from time to time. This comes from the way the insurance company generating the quote is reading reports.
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.