I'm trying to get an idea of how this campervan and campsite scenario goes. Do you have to book ahead at campsites for the night? Once in a campsite, are you not able to leave it again, as you will lose your spot? I'm really at a loss on how it all works as I've never rented a campervan before. Here in the states I see campsite and the motorhomes parked there are somewhat permanent and they tow their cars along so they have some type of transportation for their jaunts. Hopefully someone out there can explain how it all works in NZ.

Can't answer for NZ but in Australia, it depends on the campground. Some will allocate a site to you and make sure it is reserved for your use, even if you go out for the day. Others can provide a market to put on the site to let other campers know that 'your' site is occupied. If neither situation applies, it's usually safe to leave something on site (chairs, etc) when you go out. Generally other campers will respect your spot.

If you are driving a campervan, it will almost certainly be wired for 240v power. At any camping ground, there will be powered sites. You hire one of these for the night and will be directed to a particular numbered site. That one will be yours for the duration, whether or not you then go out driving. You may or may not be given a card to allow you to open boomgates into the park, and the same for a key to the amenities block. It's all pretty well organized, and you'll figure it out within a day.

As the others say - the site is yours until 10am on your day of departure, so you can unhitch and re-hitch as you like. The main question is whether you will want the nuisance of re-housing your electric cable, screwing down your vents, turning off your gas bottle, making sure all drawers are properly shut and that, generally, nothing is rattling around, before taking off just for a 5 minute ride to the shops for example. Chances are you'll only want to do it for a proper trip somewhere and that you will be happy to walk or use other transport on other occasions. This can be relevant where you've planned to eat out and the campsite is some way from the nearest restaurant. As you will have seen, the campervans in NZ are generally smaller than US RVs. Very few examples around of ones with cars hitched to the back. Good luck.

Feel free to take your campervan offsite for daytrips. We rented a campervan in 2004 in NZ, and usually took the van on daytrips and shopping trips to get groceries. Leaving a site vacant while gone was not a problem.
The campsites were excellent. The two main campsite chains (Top 10 and Kiwi) will book your next site for you from your current location. We had four people (two adults and two children), and we were glad we booked a 6 sleeper campervan for the extra room and sleeping space. Don't pay for the highest level of insurance -- we regretted paying so much.