I used to flip my old one-speed over as a kid, but got out of the habit when I switched to a 10-speed (back in the 80s) because the bike landed on the cables coming out of the brake hoods. Now I just lay the bike down on the grass. I don't think it could hurt anything.

I think I might patch, which I can do with the tire still on the bike, but the one time I tried that I had those god-awful slime filled tubes; the slime's main purpose is to spew that crap all over so the patch won't stick, so I wound up replacing the tube anyway and throwing the tube with that nasty stuff in it away. Next time I flat (only about 1-2 times a year) I'll try patching my now slime-free tubes.