Both of these are long-standing problems for me. I come back to the boards every year searching for a new solution or work-around. I also have about 30 groups in my course, and it takes way too long to figure out what group a student is in. That search feature is such a TEASE!!! I did see a user suggestion to fix this issue in this year's search, and someone had marked it as "added to theme," but didn't say which theme. I tried to find it to see if it could be upvoted/supported, but the themes list is a mile long, and "groups" isn't a theme.
I've also experienced the same problem with the mobile app, but I don't use it nearly as much as I do my desktop.