The routing looks fine, except that you have to finish in the same country you started in (with some exceptions, but Australia/NZ isn't one of them).

On your questions:

1) I think you may have misunderstood the rules: it is the total time from leaving Auckland to arriving back at Auckland which has to be at least 10 days, not each individual stop! But you should plan to speond a few days here in HK if you can - there are heaps of things to see and do.

2) LTN and LGW are quite a way out from Central London - you wouldn't want to be going between them (or to them from the centre) by cab unless you have plenty of cash. Buses and trains are the best way. There is plenty of discussion on how to get around between the London Airports in the London Forum of FlyerTalk Travel.

Hope that helps.