Great choice, I loved NZ !! I hope you're planning enough time to do both South and North Island. I'd say it's 3 weeks minimum.
highlights in the south were Milford sound of course, but also the glaciers, and in the north of the island, the beautiful Abel Tasman NP
In general you have tons of hiking opportunities, from a few minutes walk to several days.
I loved for example Roys Peak, quite easy one (a few hours) with amazing views.
For the north Island, there is of course the Tongariro Alpine Crossing, which is a great walk, easily done in day, with beautiful views
As for the gay nightlife, it's all more happening in the Northern Island with most of the countries gay bars and clubs concentrated in Wellington and Auckland
Christchurch I haven't been myself, but I've heard there are some options for gay people. But I wouldn't expect too much
Wellington is a small city, but with a lovely atmosphere. You have a few gay venues and we spent a night going out, it was really nice, we went to Scotty And Mal's and had a great time !! And in Auckland, the city is bigger so they have more gay bars and gay clubs, we decided to go to Family bar and club, it was a great choice !!
you're going to love your tip !!!!