Hi Brandi
I live in Auckland so can reassure you, weather today was lovely and sunny. A bit cooler now but no rain. Can't guarantee what it'll do over the next couple of weeks but even if it does rain, it's usually only for an hour then clears up.
As someone else said, the best spot for you to do whale watching would be Kaikoura, just north of Christchurch. Not really any whales anywhere near Auckland or Rotorua.
Ziplining, I'd recommend you do on Waiheke Island - catch the ferry from downtown Auckland to get there. Its a nice boat trip across the harbour, too. Ferry goes multiple times each day, and a lot of people commute from Waiheke to the Cbd for work each day so its not too expensive.
Would also recommend you check out some of the vineyards on Waiheke for lunch or wine tasting, too. I like Stoneyridge the best but there's a few good spots!!
Alternatively, you can do Ziplining in Rotorua at Skyline luges - it's a much shorter zip line though. The Waiheke option, you do three zip lines through the treetops and it's a real experience, whereas the rotorua one is over in about two minutes and you spend more time getting strapped in than you actually Ziplining. But if you're a bit nervous then rotorua might be an "easier" option.
Kayaking you can definitely do in Orakei, which is about ten minutes from Auckland CBD. It's nice but there may be better options near Picton? Ive only been to Picton once, to catch the interislander ferry (and we were in a rush) but i have a feeling Picton would be better for fishing too. But honestly NZ is so surrounded with water that you can go fishing pretty much anywhere!
Good luck and feel free to pm me if you have any questions :)