If they are GTC (Good 'til canceled) and renew automatically they will no show up in completed listings as unsold or even as sold if a multi-qty listing and some of then did sell.
https://community.ebay.com/t5/Technical-Issues/eBay-please-fix-this-exploit-some-sellers-are-using-t...
If the plain old "relist this item N times" auto relist then they should show up in completed items.
Sounds like that item could be a GTC item that is "ghosted" and doesn't appear in completed items.
So no, relisted listings are no indication the seller is active. A seller with eBay autopayments enabled (to cover store, listing fees, any FVF's on eBay invoice, etc) and some GTC listings could go walkabout for a year with the store or listings on autopilot, and look active as long as nothing or little sells (if something sells, seller on walkabout doesn't ship, seller metrics get hit, eventually account is restricted if happens enough)
Not sure what you can do if you tried messaging with no response. Maybe direct email? (you must have seller's email address from PayPal transaction details or PayPal emails somewhere if you purchased from him in the past)
eBay removed ability to request member info (like phone number) a while ago and I don't thing there is any workaround or backdoor for that.
Maybe social networking (Facebook, Twitter, LinkedIn, etc) if the seller had a presence there and you engage in that sort of thing.
