1) Yes exactly that. If there's nothing abnormal about the stairs (going around turn, small treads to step on, other obstacles, etc) just walk down the stairs and see if you turn your knees in any direction other than straight forward (in the direction you are heading on the stairs). I was told (and did no other research) by the PT this was a normal reaction for people who didn't have great knee stability.
She did some tests and decided it was weak hips for me, specifically the abductors (I think... the muscles you use if you stand up straight and lift your leg up straight out to the side, 90 degrees off from the direction you face. Lazy on the googles, sorry) There were some simple tests the PT used and yours should be able to do the same, I think. The story for me was weak hips led to knee instability, = knee pain, but also led to me somehow compensating with my calves which made them super tight. No idea if it works for you but can't hurt to ask. I thought I was in great shape at the time, playing soccer, ultimate frisbee, hiking, cycling, running, etc all the time. Ends up none of those strengthen those hips apparently.
2) I had to go look through orders, but there are a ton of options on sleeves. PT might have recommendations, but I found generally the tighter the better. I have a pair of https://www.amazon.com/gp/product/B00D8GOR66/ and https://www.amazon.com/gp/product/B00EBHPP3U/ but they definitely weren't $35 when I bought them. I liked the CEP ones slightly better but they worked about the same. I would wear them while playing soccer or running sometimes and that helped, but definitely would put them on afterward. There are varieties with socks, and you could probably wear them hiking (like tall soccer socks) if they help you