The only place where an AR is needed on this arrangement is on the turntable bridge, and you need something there anyway. I'm not sure how an Atlas turntable is wired, so you may need to gap both rails on all of the tracks leading into it. Short stub tracks that don't connect to anything else should be OK. This assumes that you aren't running trains across the turntable bridge.
One point on terminology, what you have labeled as a crossover is actually a crossing. A crossover is a pair a turnouts put in so that you can move across from one parallel track to another.
The crossing shouldn't be any problem. These are all made with insulated frogs so that each route through the crossing is completely independent from the other.