I host international (non-USA ) guests. Actually, most of my guests are international, whether they are students or here as a tourist. I'm also well versed on the American visa processes. It's up to them to figure out how to get a visa. My only exception for listing me are international students, where they are required to input where they are staying at time of entry. Even though they don't ask, I allow them to use my address as a temporary stay (all international visitors are temporary).
1) I'm not familiar with New Zealand immigration and visa regulations. But I do know that it's not up to me(host) to help them obtain a visa. They should be able to get that separate from you. They may be asking because their visa request is weak.
2) You have a lease agreement with your landlord to live in your home. I'm assuming it's an apartment. Any guests that stay with you do not, unless you want to live with them on a permanent basis. Guests have a 3rd party agreement to temporarily stay in your home. No more.
If I had a guest ask me to give them a rental agreement just so they can get a visa, I'd suggest they better get a hotel for a month and ask the hotel manager for that agreement 😉
Does this help you?