Thanks to all the great advice I've gotten here, I'm now considering some very interesting itineraries. Pending availability, of course, is it possible I could book three one-way coach segments, something like the following?

LAX -> MIA (stopover) -> GIG on LAN for 20k

GIG -> BCN (stopover) -> JNB on IB for 35k

JNB -> HKK (stopover) -> LAX on CX for 45k

Seems too good to be true as is, but is there any chance of getting more than one stopover on a single one-way segment, e.g. LAX -> NYC (stopover) -> MIA (stopover) -> GIG ?

If I want a simple round-trip ticket, could I just book something like LAX <-> JNB on CX, avoiding the BA fees? If so, how much might the fees be for that route on CX? They are $1100 or so on BA metal, and I can't get the CX route to show up on BA.com's search because BA always has availability on that route.

Thanks again everyone. Can't tell you how much I appreciate the help.