Everyone does it a little different. For me, drainage is a big concern since we get a lot of rain and the run isn't covered, so the base of my litter are chunky wood chips, chipped from branches in our yard and left to age 6 months+. What I add after that depends on the season: fall and winter, dried leaves (we rake them up and store them in the greenhouse, I try to collect enough to last through spring). Spring, some dried leaves and grass clippings, cut short and left to dry. Summer is mostly grass clippings. I add some garden trimmings and weed clumps too when they're available. A little bit of store-bought pine shavings get mixed in too, only because I use that in my nest box and I clean those out when I clean the coop (and I use the same chunky wood chips in the coop, so coop litter gets cleaned out into the run and becomes part of the run litter).
The run gets cleaned out when the litter has broken down enough that it's not draining well any longer. That happens maybe once a year, and I only clear out the stuff that's pretty well broken down, and that becomes compost for the garden beds. Then I add more wood chips and the process goes on.
You don't have to remove poop, though I do use a pooper scooper and pick up big obvious ones daily (which go into compost... everything chicken is compost!) just so I don't step on them.
As far as flies and smell, if you do this correctly, there should be no noticeable smell from the run. Flies will always get attracted to fresh poop though, but I put up fly traps so the fly problem has been minimal... I probably get more flies around my house than the coop!
