For the judgments......yes, send the county clerk at the courthouse the letter asking for their procedures in verifying records.
You can send a
PR (procedural request) letter regarding the medical......but I've just never seen that letter do any good except for giving yourself a papertrail if you need to send an ITS or decide to file suit later on. You can try that......I just haven't seen much success with it.
For medical accounts that old.......my advice would be to either DV the CA or contact the healthcare provider and ask them for proof of service.