I have the Children of John and Emma. I have everything for the next generation down, they are not the tricky points.
As for the Carl Carlson... it 'might' be him, however, it also might not be... I have no way of confirming that, and bdm seems to be of no help as well as ancestry, and FindMyPast.
The one thing that might help is that John died in 1905, and in in affadavit Emma signed in 1907 (very delayed) it stated he had no relatives in the colony of New Zealand. This must mean that either the parents (Carl and Carolina Bjornvik) died prior to 1905/7 or moved from New Zealand prior to then.
I have no idea what ship they came over on, and if I was able to find that might have a little more luck with sourcing the Bjornvik information (as far as I can tell the Bjornvik name is non-patronymic, which is unusual for Finland)