Having a large family in terms of size as well as numbers plus bikes and a dog to fit in, you are going to require a fairly large vehicle.
Engine size will also need to be around 2- and 2.5-litre especially when you are fully loaded, so you'll compromise on fuel economy.
I wonder if you should think about having the bikes mounted on the roof of the vehicle. If your son is 6-foot plus, loading and unloading from the roof should not be a problem!
You may also want to budget for the installation of a barrier between passengers and the dog for safety reasons. If it's a large and heavy dog it needs to be secured in its own area for its own safety as well as that of other occupants.
In this price range safety features should include multiple airbags but also try and find makes/models fitted with electronic stability control.
Subaru Forester X 2008
Plenty of power from the 2.5l engine plus it has the off-road ability if your cycling journeys are going to take you into mountain bike territory or you want to get into the extreme outback with the dog and family.
The 'X' variant is an NZ new model which I would recommend. All NZ new Subarus have what is called datadot security which means they are far less likely to be targeted by car thieves in comparison to similar aged used Subaru imports.
Honda Odyssey 2007
The 2.4l Odyssey has loads of interior room. They are a seven-seater but if you fold the third row down it opens up plenty of cargo space which the dog may appreciate. It's not really designed for the extreme outdoors due to its reasonably low ground clearance.
Mazda6 GLX Station Wagon 2010
Safe, reliable and comes with the smaller 2.0 litre engine so fuel consumption may be better depending on loads carried and the type of terrain you intend travelling over. In their day, they were a popular choice for business reps that were on the road constantly.
Driven recommends
Past service history and trying to keep the odometer reading as low as possible will be important once you have selected the make and model that best suits your needs if your intention is long-term ownership of the car. Regardless of where or how you decide to place the bikes on the vehicle, make sure you check out costs.