I have read the rules and tried to use the on-line planner but am still not sure the following is a legal routing.

DCA-MIA-EZE then EZE-LIM-LAX (stopover)-LHR-CPT...HKG-ORD-DCA

The rules.

(e) Only one intercontinental departure and one intercontinental arrival permitted in each continent except
as follows:
1. Two permitted in North America when one is a transfer without stopover.

(f) Only one international departure and one international arrival from/to the country of origin permitted
EXCEPTION: Two permitted for origin USA when one arrival-departure is a transfer without
stopover.

If DCA-MIA-EXE is without stopover then can I come back into North America with a stopover in Los Angeles?

Thanks for any help.

It has been a while since my last OW RTW but i believe the first (DCA-stops in the US-EZE-xLAX-LHR....DCA is valid (x stands for stop).
The second example (stops after reentering North America) breaks the rule, since you don't have what is mentioned in the exception.

If you start outside of North America you could do the transfer either at the first or the second time in north America.

The rule you quoted above is for intercontinental departure and arrival.

The rule you need to read is the one on stopovers, specifically #2 below:

Bottom line for your itinerary that starts in NA, is you can do a stopover on the way out of NA on your initial departure from that continent (in your case for DCA-MIA-EZE would mean a stopover in MIA) and then another stopover on your final return to NA at the end of your RT trip (in your case HKG-ORD-DCA would mean a stopover in ORD). You cannot do a stopover in LAX on the transit without stopover (LIM-LAX-LHR) portion of your itinerary (that's the "without stopover" part) - you have already left NA which means you can't return there until the end of your RTW trip. They allow you a transit without stopover so you may get to your next continent.

Your quote "DCA-MIA-EXE is without stopover" is not a transfer. The full context of the rule you used out on context is transfer without stopover. DCA was an origination, not a transfer. Your EZE-LAX-LHR is and has to be a transfer without stopover in order for it to be allowed.

Since you're going to JNB anyway, it might well be cheaper to start there, even considering the positioning flight, depending on several factors.