On this note, I use spreadsheets for planning. I use Expert Flyer to identify flights that look enjoyable, and put each flight in one row of the spreadsheet, with columns for from airport, to airport, flight number, booking inventory, departure time, arrival time, arrival day offset (0, +1, or +2 for, e.g., LHR-SYD), flight duration, flight miles (all of which I get from EF). I have columns that calculate the day of the week (helpful for planning, as some things we may want to do and some flights are only available on certain days of the week) and the duration between flights, and the earnings. I leave a blank row for a stop (the duration between flights calculation skips over a blank row).

The Tour Approvals Officer is the spreadsheet specialist in the family and I have passed your suggestions to her. 😉

We really like the order of the Asian cities and Australia/NZ in pandaperths plan.

Would it be allowable to route AKL>HKG>CPT(via DOH)>OSL(via LHR or DOH) as the tool says this is against the rules, 2 stops in Europe or Middle East not allowed if traveling to South Africa? I can only see Europe mentioned on the OneWorld website?

There's a restriction in the rules regarding stopping in Europe twice to/from South (another reason why an EF subscription is so handy: it provides fares and fare rules):

So I think you might be OK if you went to South Africa to or from Asia or Australia one way, and Europe the other.

Thanks.

That’s how I read it, so I was trying AKL>HKG>CPT(via DOH)>LHR>OSL. I was thinking of breaking the trip at LHR and doing a short trip to Oslo later in the year with the last segment, but the tool does not like it!?

How many stopovers will you already have in EUR/ME up to that point?

None.

Plan is to start in OSL, then to NYC via LHR but no stopovers, just a connection.

What you want to do is allowable.
Two of the rules of the ticket are in play here:

Rule 8 StopoversSecond Visits to a Continent - Rule 4(e)NOTES
The online tool is very buggy, so never takes its word when it says an itinerary is invalid! (But I've just plugged your itinerary into the tool and it says it's valid!)

The full fare rules are available on the Oneworld web site, the User Guide tells you how to find them.

Provided you haven't stopped over twice in Doha (!), you can stopover in London after Africa
(or before Africa, if you fly HKG-LHR-CPT-DOH-OSL, or both before and after Africa if you don't stopover in Doha at all and you fly HKG-LHR-CPT-DOH-LHR-OSL :D).

In your first post you asked about the optimum time to book. I don't think there is one. As Mwenenzi posted, 'D' space can sometimes be hard to get, so booking earlier rather than later might be indicated - however my two most recent RTWs I booked 6 days and 10 days before their first flights without any problems.
ETA:
A reason for booking earlier than later in these inflationary times is the possibility of fares increases. (For interest, I checked ex-Norway fares, and they are cheaper now than they were back in 2012 !!!)

Qatar Airways business class is far superior to British Airways business class (OSL-DOH-JFK)

Need to write out the complete route and check against the rules. AKL-HKG-CPT-DOH-LHR-OSL [the "via" will trap you some time]
QR also fly into other UK airports. Not just LHR
A map from the Great Circle Mapper - Great Circle Mapper

Ah, looking at the map you posted I realise I had totally forgotten that CX flies to South Africa from Hong Kong! Another option for AlanT :D
Checking Expert Flyer it looks like CX has not yet returned to Cape Town, but it is flying to JNB

ETA:
On second thought, this is not an option - because of the rule regarding a second visit to Asia, which is
"Two permitted in Asia when one is for travel between the Southwest Pacific and Europe/Middle East."
AlanT's first visit will be from Nth America to SWP, and the second from SWP to Africa - so not allowed.

Yes, far better. I suggested OSL-DOH-LAX-JFK rather than OSL-LHR-JFK, but the important point is the much nicer (and longer) QR flights. OSL-DOH is likely a 1-2-1 configuration, much nicer than BA Club Europe, and DOH-LAX or DOH-JFK should be Qsuite, arguably the nicest business class in the air (certainly the nicest in OneWorld).

We took OSL-DOH on QR a couple of weeks ago and I agree. By far the best business service we have had. Attentive (almost as much as JAL aim a good day) and very nice seats even on the 380 which is supposed to be less good.

Due to a medical problem we reached OSL checkin 60 seconds after the desk closed and the local agency staff were absolutely horrible: cold, unwilling to help, refused to call the QR manager. Eventually we did get them to consult QR and the manager was exceptionally helpful, reassuring and professional; he even came down on a scooter to reassure us as we rushed from security to the gate (which was, as always happens when things go awry, the very most distant one). Huge bonus points to QR but not their local Norwegian agent sadly.

We have moved QR to the top of our favourite list.

Having just said how much I like QR , that made me think: if I go OSL-LHR-xDOH-SYD then US then LHR-OSL, is it allowed to do the US-UK bit back through DOH on QR? I think we are allowed to go through DOH twice because one is a direct flight to S Pacific, but would it violate the continuous TC1-2-3 rule?

Yes
No, Middle East is part of Europe.

There is no limitation on how many times you can go through DOH, the only limitation that might cause some issues is you can't fly the same city pair in the same direction more than once. Also, since Europe is your originating continent you are only allowed two stopovers in Europe.

I always end up going through DOH multiple times on my regular RTWs
My record is four times, back in 2015
JNB-xDOH-EZE......BKK-xDOH-CMN-xDOH-OSL-xDOH-NBO :D
As anabolism said, there is no restriction on the number of stopovers/transits you can have at any point (other than the point of origin - no stopovers or transits allowed there)