Could this S trap be causing sewer gas odor and drain backup?
Oh my. this trap isn't even an S trap, it's a rollarcoster trap. wow..funny, but not your issue. The reason for the P trap is to trap a little bit of water in the P where stop all air from moving up the pipe. You have a sewage backup issue. 

It's very possible this trap also does not trap water if you take off the trap you can see for yourself if it's full of water or not. The suction that S type traps create pull all the water out of the trap like a siphon leaving no water left to block air flow. I have one, i need to get around to fixing. It definitely does not cause any backup.