I am British & my wife is Japanese. Our daughter was born just about 6 months ago.
From my brief study of citizneship I understand that our daughter may take dual citizenship of Japan and UK until she is 18. From there the British system still recognises her has British - the Japanese system requires her to choose between the two. I think practically as long as she doesn't advise the Japanese authorities that she holds a British passport after 18, then she can get away with both.
Whichever way, her children do NOT automatically get the right to British citizenship. I have not gone into the details of how her children might get British citizenship if they wanted.
As for New Zealand, I have no real idea though from kiwi friends I have I seem to remember that the kiwis are pretty relaxed on dual citizenship.
I suggest you pop into both embassies to find out more details - they were quite helpful in Japan when I asked at the British Embassy.
Registration is very easy - just go to the British Embassy in Tanglin and you can register your child and if you want also get a new birth certificate. Do remember that you need to obtain a dependents pass from the Singaporean immigration (which we forgot for a while!)
As for your other posting, my wife had a not-so-bad pregnancy and things were quite straight forward from 4-5 months onwards though different women seem to react differently. My wife made sure she had plenty of rest, sleep etc and that seems to help. There is a good resource in Tanglin Shopping Centre (4/F) called "Childbirth Education" run by Anna Wong. It operates different classes that we found invaluable and it was good to share our experiences with others going through pregnancy.