Have you thought about running shoes? Super-lightweight, very supportive, well-ventilated, made for very high impact....
My current pair are Brooks Ghost 7s, which I love... but I have extraordinarily picky feet and require very odd features (like overlays that don't affect my bunions). If you have an easy-to-fit foot, any running shoe (that fits you well & feels comfortable) should work.
I don't generally wear them for my indoor workouts, because I sized up a full size for running (as recommended), and they feel strange on carpet--a bit like clown shoes! LOL But if I'd bought my regular size I bet I'd love 'em.