Looks like the rings are designed specifically for this kind of work?

Once each one's ends are joined together, do they need to be glued or welded or something?

If you were actually going for historic accuracy, or planned to use it for armor then you would weld the rings. Generally people don't weld them since it takes a lot of time and effort. They are just bent closed.

Historically it would be riveted not welded. Historical example

Correct. Occasionally in my darker moments I've actually considered making a set with riveted rings. Every time I've ultimately come to my senses and concluded that I can't be bothered.

You'll notice that authentic mail usually uses rings that are flat in cross section, too, what with medieval Celts or whoever not having access to unlimited amounts of our modern machine made wire.

I can feel the amount of time and effort on display in that photo.

I used to have a friend who would make chainmail. He had long coils of steel which looked like springs but I don't think they were actually springy. He'd nip off individual rings from them, weave them into whatever he was currently making, then bend them closed.

When I was a kid, soda pop cans had tearaway tops that would separate from the cans when opened. There was sort of a ring-shape on one end of that piece. SCA people would collect those by the hundreds or thousands and use them to make mail armor suits and stuff. It looked surprisingly good, and since it was just thin aluminum, was very light. Very good for running around on a hot summer day recreating the Battle of Hastings or whatnot.

"Designed" is a strong word. I crank these out, literally, on a wire winding jig that I made out of scrap wood and bits of steel rod a couple of decades ago. Since these are aluminum and I wouldn't drive myself bats doing so, I cut the rings off of the coils using a jeweler's saw. (You can do this with steel wire if you really want to, but it takes forever and you'll burn through a lot of saw blades. Most people just use wire cutters to make steel rings, which are uglier but the process consumes significantly less of your life.)

Thanks for sharing your process!