First, to get a stuck freewheel off I put the remover on the freewheel, use the skewer to secure it (not completely tight but very close), hold the wheel perpendicular against a wall, put a 10" crescent wrench on the remover and give it a couple of solid whacks with a rubber mallet. This hasn't let me down yet even on 30+ year old neglected freewheels.
As for the 2 races, I believe what you are seeing is the outer shield and the race. Looking through the freewheel they can look similar but they aren't. The bearings go further in than that. You will know they are in the wrong place if you assmble the cones, spacers and nuts on that side and you run out of axle.