Yes, you will need an NZeta visa. This is very simple to get (I recommend downloading the app, as it makes it very fast and easy). Note that you will need one visa per person and please allow at least 72 hours for processing, so do not apply the night before you leave or at the airport! For more information: https://nzeta.immigration.govt.nz/

Yes, a car is highly recommended. NZ public transport is nearly non-existent and you will need a car if you are planning to do some day trips or even see the sights locally in Queenstown. If you book in advance, rental cars are not expensive. Note: vehicles drive on the left hand side of the road! And there are many tourists that don't know how that works, especially in Queenstown, so please be cautious of others when on the road.

We recommend Pak n Save on Hawthorne Drive - it is one of the larger grocery stores: https://maps.app.goo.gl/BHapC6BCAfpa6RW8A