I sometimes run on my paths and on Main Street, but I'm trying not to because one time I was running in Main Street with a balloon, and I found out I had bad luck that day when I tripped and lost my balloon... then I got another one out my drawer, forgot that I had bad luck, ran on one of my paths and lost my second balloon that day.
Nope, I never run in my town as long as I can help it (except for in the shopping mall, inside buildings, on the beach, and on the stone areas outside the plaza and buildings). I'm super paranoid if I run over an area just once, it'll kill off part of the grass. Awhile back, I thought patterns protected the grass, so being the idiot I was, I made paths and ran on them 24/7. It felt nice, since I'm impatient, and walking quickly is pretty slow compared to running. In the end I researched about running and grass deterioration, and I've not ran in my town much since then. Hopefully in the next game they have an option to turn grass deterioration on/off.
I only run on the paths. I know my town layout so well by now that I can run without worrying about accidentally getting off of the paths and destroying flowers.
in my first town i have paths, so yes, definitely. my second town doesn't have anything down yet so i try not to in that one, but i usually fail because i hate walking so slow ;;