Hi there,
How can I build a new website and then transfer the old one over?
You should be able to find lots of articles on Google for migrating sites/themes as it’s not a theme-specific question.
I would say try starting a staging site, duplicate the existing site over, activate GP, make the tweaks, duplicate the site over to the live site once you are ready.
(Also, the website is very high ranking, number 1 for the majority of it’s keywords. Will transferring the website to a new theme affect the rankings negatively?)
As far as I understand, ranking is based on URL. So if your URL doesn’t change, then it should be fine.
Worth doing some research on this as well though 🙂