Yes - three times through DOH is allowed.

In general, there is no limit on the number of stopovers/transits at any point. (The only exception is the limit of two stopovers in the continent of origin).

Your itinerary is valid, and if the booking tool allowed the first flight to be on QR, then you could no doubt book it online. An option would be to add-on say DUR-CPT at the beginning and then it could be booked online (BA would be the first carrier); similarly you could change CPT-DOH to CPT-JNB-DOH.

Which airline are you planning to contact to get this ticketed?
It might be useful for you to read the first post in this thread http://www.flyertalk.com/forum/onewo...cpt-done5.html

Booking through a carrier might be an issue
The issue is that AA (and until recently BA) misinterprets the rule regarding the second visit to Europe/Middle East and would disallow your DOH-LHR and LHR-DOH flights "because inter-zone flights are not allowed". See from post#73 onwards in the thread.
Long story short, I finally got the BA fares team to change its interpretation and allow inter-zone flights.

Someone () now needs to get AA to change its mind.
Please do let us know how you go if that someone is you