Except for the few times of year tht I'm fishing for steelhead, I only carry dry flies. That simplifies things a lot for me, although I can still be pretty loaded down with boxes during the month of may, when you'll liable to see anythign hatching.
But i move flies in and out of my boxes according to the season . In april, I have all my blue quill, quill gordon, hendrickson, and grannoms etc. First of week of may, most of them go, and are replaced with march brown, gray fox, sulpher, and green drake
Now, i'm ready to move most of them out, although I carry sulphers year round - I've even seen them hatching in fall.
Besides several boxes for mayfly duns, I have a whole box with nothing but spinners, and another with caddis only.
Also separate boxes for terrestials and small stuff - flies under #20.
As for box type, I use just a standard, 8 compartment clear box - no ripple foam. I've found that keeping dries in these type of boxes messes up the hackle - after a while it gets bowed out.
I tried the chest boxes when i first started fly fishing 25 years ago - didn't like them. They just seem to be in the way - especially when landing and releasing fish.