Just had to pop in and say hi xD
Using a mirror to encourage him to flare is perfectly fine and gives him some exercise, it's not too much stress as long as you don't over do it, like leaving a mirror next to his tank or making him flare for long periods of time. Just use common sense. :)
Personally, for a 1/2 gal, I would suggest doing a full water change at least twice a week (every 3 days should be okay), I don't really find partials necessary for such small amounts of water. A less stressful way of doing changes is by "cupping" your betta with a small plastic cup (I use the one mine came in) do the change, and acclimate him back into the water.
Don't worry, all of us experience a learning curve where we learn what bettas really need vs. what we've been told, I certainly did, and now my boy has a 5.5 gal NPT with ghosties and a heater xD I spoil him (that and I'm lazy and don't like doing a ton of water changes, been there, done that)
Filters are not very important with bettas and are completely optional, however, if you want anything bigger than a 5 gal, you might want to get one if you want other fish and don't do live plants (100% changes on large tanks are a royal pain)
Hopefully you get him back soon :) best of luck with your buddy!
5 bettas: Шостакович Shostakovich, Dimitri, Blood, Fancy Pants & Miss Priss; 1 cat: Rajah