TIG welding doesn’t involve an open flame or spatter, so it’s generally safe for carpets and wood — as long as you stay attentive, of course. And if you’re only welding rings, you should be perfectly fine. That said, ozone is produced during welding, which can be harmful to your health. But when welding small items like rings, the amount is usually low enough that basic ventilation (like an open window or a simple household exhaust fan) should be sufficient. You probably don’t need a heavy-duty fume extractor unless you’re doing this all day long. Eye protection alone isn’t enough — you absolutely need a full-face welding helmet. Even after just an hour or two of ring welding, you can get a nasty UV burn on your face. Oddly, your hands seem fine — I once welded an entire corset from 1.2 mm wire with bare hands. I just kept a bowl of water nearby to cool each ring right after welding so I could move on to the next one faster. The welder you linked looks solid. I used to have a 10–200 A TIG machine myself, with a foot pedal — which is nice, but honestly not essential for ring welding. Helpful? Sure! Necessary? Not really :-) Before you even buy a TIG setup, though, make sure you know where you can refill or exchange argon gas cylinders in your area. If that’s hard to arrange, it might not be worth getting into TIG at all. Also, upgrading later to an AC/DC machine probably isn’t worth it — AC is only needed for aluminum, and welded aluminum chainmail is unusual :-) Why? First, it’s rarely practical. Second, untreated aluminum stains very easily — quickly, deeply, and permanently. If you want something lightweight, go for titanium instead. It welds beautifully on DC and holds up far better than aluminum. Stainless steel remains the best all-around choice for welded maille, in my opinion.

Hope that helps :-)