tkayed:kenkeniff: Do they give you an option in their control panel to add more domains or sub-domains to your site?
no. only the site address, which is hypnobirthing.kiwi.nzkenkeniff: In that case your best bet is to find a DNS host that does URL forwarding for individual sub-domains and setup the ones you need as previously described (so you don't need to rely on the random Geekzoner's servers being up 24/7).
do you or anyone else recommend or even know of any NZ-based services that I can use for the DNS host? And if so, does that mean I need to point nameservers and then setup DNS zones with that provider, correct?
Yes correct, don't use any dns hosts like that myself so can't say.
Maybe find out if all your subdomains pointed at Weebly are supposed to just work anyway as they still maybe able to diagnose their issue using www-test...