As you have been told, you can apply, and link your application to the spouse's student visa by giving all details in a covering letter and include a copy of the AIP message.
1. It depends on how things are going in the INZ branch you have to apply to. Use Google, and enter "Immigration NZ (whatever) branch processing times." The page that comes up will give you that branch's own estimate of how soon each kind of visa can be expected to go through. But notice, INZ timings are never a promise. For one thing, every case is individual and has to be checked out, and for another, the timings are just their best guess based on recent averages. It only takes a sudden flood of applications just before yours to make the estimates meaningless.
2. As long as you have good evidence of partnership, and you meet the other requirements, the visa should be granted. It is entirely dependent upon meeting the requirements.