You're right, Christchurch is in Canterbury - but you didn't mention either in your first post. No matter, we've got it now. :-)
If you go back to the birth fiche you were looking at, you should find a District Key that will tell you what office any particular folio number was registered at that year. This might help you to narrow down some of those children to just a few where Henry and Dorothy were living at that time. Or even only one if you're lucky!
Sometimes you'll find the booklets produced by the New Zealand Society of Genealogists (NZSG, http://www.genealogy.org.nz/
) alongside the fiche, which I've found the easiest way to look a number up - as long as you make sure you look at the right year.
Alternatively, the lists should be on the microfiche themselves. Probably either at the very beginning, before the A surnames, or at the very end after the Z surnames.
I know nothing about vaccination certificates, but divorce files go to Archives New Zealand (http://www.archives.govt.nz/
) when the courts are finished with them. There's usually a 60-year withholding period for divorce files.
Their online index is Archway (http://archway.archives.govt.nz/
), though be warned it is not complete - sometimes you may need to write to Archvies and ask if they have a file you're looking for.
It is free to walk into an office of Archives NZ and view files that are open-access. You have to go to the right office, they're not moved around from office to office for your viewing convenience. Or you can ask a friend to visit for you and take notes or photos, or you can pay Archives to send a copy of the file to you.
I've taken a look at Archway just now, and can see the index entry for the divorce file for Henry and Dorothy. It's marked in red at the side as having restricted access - click on the "order details" link to find out more about the restrictions. I think it unlikely for you to get access to this file before it comes off that restriction in 2021, though Archives staff can advise you more accurately.
Newspapers might be another way to go for information about the divorce. There used to be a paper called NZ Truth (might still be for all I know!) which frequently reported on things like divorce cases. PapersPast (http://paperspast.natlib.govt.nz/
) carries their editions through to 1930 - too early for you for this particular event, but I mention it in case it's useful for something else. You may be able to find that paper in a local archive, or perhaps National Library of New Zealand (http://www.natlib.govt.nz/
) can help you access the 1961 copies.
It might be that someone reading this board is living in Christchurch and is able to check out some local repositories for you. I'm not sure what's available there at the moment, with the chaos caused by the earthquakes (more than 10,000 of them since September 2010!), but I know Archives NZ is open (though some files are off-site) and the central library has reopened in a new location - though whether that includes their fantastic collection of parish records I don't know.
Give it a little time, someone may be able to help you but perhaps just not this week. I only check the boards daily because I'm the admin for some (like this one) and like to keep on top of my duties, which means reading all posts on "my" boards.
Hope this helps. :-)