We will be in New Zealand in April, the adventure capital of the world, but supposedly I can't do anything because I'm 5 months pregnant. I can't hike (was planning on doing the tongariro crossing and milford track), I can't abseil, I can't ski, I can't parasail, I definitely can't bungy jump, I can't even kayak or jetski. Is there anything that I can actually safely do without risking harm to the baby? Baby comes first, even if it means that we'll just be like an elderly couple going on sightseeing cruises and taking photos sigh.
Who says you can't hike?
If you have no medical concerns (ie high blood pressure, blood results showing complications etc) then guidance is to keep doing the sports you enjoy (with some exceptions - generally sports where impact or similar forces could cause injury - bungee jumping, motor racing etc - and those that could cause excess damage to ligaments, as these soften towards the later stages of pregnancy)
I don't know the two particular hikes you mention, but as long as you are aware of your water/food requirements and plan accordingly, there is nothing to stop you hiking as you normally would.
Kayaking may depend on the water - I wouldn't plan an extreme white water trip, but lake or calm seas or rivers may be fine - plan for more frequent stops and remain comfortable.
Skiing is similar - don't plan on black runs, but you'll find many people ski on more moderate runs while pregnant. If you feel like you are comfortable doing it at 5 months then just do it. My wife was still running 10k runs at 8 months with one of our children, and did a 10k walk the day before my youngest was born.
Keeping fit and having regular exercise is considered a very good thing during pregnancy, so while you won't manage adrenaline sports, activities in general will be fine.
Plus: It's your honeymoon - have fun!
It depends on where in NZ you are planing to stay. If the first idea was to enjoy outdoor sports, this may be compromised...
Whale watching is a possibility, although the tours are done in rough weather as well.
Visit a sheep farm, go to a marae, have your man participate in the haka and film him for your kids ;). Visit a winery, the whites are famous and often fetch top prizes (many swiss people work there, is this a certificate of quality?). Don't gulp the glass down, delicately sip and add air in your mouth to savour flavours and fragances and then spit everything out, as connaisseurs do.
You can safely sail in the Bay of Plenty or stay in a hotel or nice Bed & Breakfast around there. Make a day trip to Cape Reinga and 90 miles beach in a confortable bus (don't be fooled, Cape Reinga is not the most northern cape, and 90 miles beach is much shorter!). Discover Napier, the art deco city and look at its history (careful for the tremor reproduction in the museum). Lake Taupo and surroundings, Rotorua and the mud pools & geyser park (prepare to smell sulfur). Stop at the French bakery there and enjoy!
Careful about hotsprings, you can go there but don't get water in the nose (http://www.cdc.gov/meningitis/parasitic.html), plus depending on your condition, hot water tend to induce birth...
NZ is known for its landscape and outdoor activities, but there are other cultural things that are possible and enjoyable in your condition! Being pregnant is not a disease. Think about other things you may like such as Paintings, History, Plants (beautiful botanical gardens like the one in Wellington - 26 ha free to roam), Scenery (Mount Vic in Wellington again, accessible from downtown), the huge sea aquarium in the old sewage station of Auckland, the national archives near the beehive. Stroll beaches and think if the movie the piano could have been shot there as well.
Depending on where you are from, and if it is not too late, think about staying a few days in Australia. Sydney is a nice place.
Plus, will you rent a car? In that case rent locally, it is usually more affordable and the quality of the service excellent. Will you fly around? Which places? Where from? Carefull about road naming. A highway can be a dirt road ;) with no gas stations... and villages so tiny that if you blink you miss them :). this is particularly true around the Taranaki, which is majestic!
One last remark... Are you prepared to feel the ground shake? Have you revised what to do in case of an earthquake? What aftershocks are etc...? I went there several times with small kids and never had any issue, but it is like Japan. Seismograps continuously pick up tremors. Don't worry, life would be so dull without a bit of risk!
I used to know people who went skydiving into their third trimester. The only reason they stopped was because sitting in the airplane was rather uncomfortable. Admittedly they were experienced jumpers - we declined to accept students or tandem passengers who were visibly pregnant but if you've been doing it for 10 years....
Abseiling: if the harness fits...
Parasailing (Mexican skydiving): ditto. It doesn't knock you around.
Jetski: take it easy and it's no different than a car.
Kayak: a) do you fit? b) avoid the rapids.
Bungee: yeah, I'd skip this one.
Skiing: done by pregnant women daily. Avoid moguls, olympic-grade jumps etc.
Or you could just stick to the more common honeymoon activities.
Well, it seems wise not to do anything that could make you fall on your bottom, although I have the impression your hubby thinks pregnant women are made of sugar. That's not uncommon at all, BTW.
Anyway, the Waimangu Volcanic Valley could be what you need. New Zealand has the largest hot spring in the world, the Frying Pan Lake. There are several attractions, this is just the most prominent. They are all along Waimangu walking track, which is wheelchair-friendly. It is a very scenic location, with lots of exotic fauna and flora.
(I live in New Zealand, Have been everywhere from Cape Reinga to Bluff, Climbed and hiked extensively in the back country and done many of the Great Walks, and many more other trips. I also have done or do MTB, Ski, Ski Tour, fish, hunt....)
New Zealand is seen as an action adventure capital. However when you arrive here you will find a vast majority of tourists are not adrenalin junkies, and even most of the junkies spend most of their time doing spectacular, but far from Adrenalin junkie stuff that any pregnant woman could do safely. Our industry has options for everyone of every physical ability.
The question is not what can you do -that is impossible to answer and way to broad. Plan your trip like you were not pregnant, tell us what you can not, or are not prepared to do as a result of your condition and I will suggest something else equally spectacular.