I have noticed that their multistop itinerary builder is very buggy, indeed. I've never managed to get it to work.
Re: stopovers
How many stopover options can I choose from?
When travelling to New Zealand and/or Australia you can choose from one of 5 Asia stopover points (Hong Kong, Shanghai*, Beijing* ,Tokyo*, Osaka*, ); one of 4 North American stopover points (Los Angeles, San Francisco*, Vancouver* or Honolulu*) and one of 5 Pacific Island** stopover points (The Cook Islands, Samoa, Tonga, Fiji* or Tahiti*).
* Some flights operated by one of our partner airlines of Lufthansa, United Airlines, Swiss International, Austrian Airlines, KLM, Air Pacific or Air Tahiti Nui to fly to these stopover destinations
** Up to an additional €300 per stopover on the base fare
So, it would appear that you can only have one Asian and one North American stopover for free at those named points (so maximum permitted mileage doesn't come into it - if you want to stitch together shorter legs, you're probably going to need a Star Alliance RTW ticket). Codeshares (with the above mentioned airlines) are fine.
To choose specific fare buckets, you'd definitely have to speak to an agent.