Clarification from original poster:
1) QF will be the check-in carrier in Christchurch (I find QF hit-or-miss on customer service in these types of areas);
2) I set up the ex-JNB DONE05 with the US AA RTW desk then had an Egyptian T/A ticket;
3) Thanks - I printed out that 5-1/2 year old PDF regarding thru-checking. At least it's something to have in hand.
4) If I can't check luggage through to LAX, maybe I'll check it to JNB. Then I can not pick it up upon arrival, stay at the in-airport hotel & pick it up later that afternoon prior to my evening check-in for RTW#2. Or just pick it up, clear customs & stay at that landside airport hotel that you can walk to. Just I won't know which hotel to book in JNB until I check-in at CHC