Hi gstyl,
Thank Mch’s involvement on this case first.
As I understand, you want to move website pages to Office 365 when adding a custom domain to Office 365.
For the first question:
Moving website hosting won’t move the web page contents automatically. As Mch mentioned, you need to rebuild the website if you want to host web pages on Office 365.
As for the second question:
Yes, you can add custom domain to Office 365 and continue to host your public-facing website with your current hosting provider. For more information, you can refer to Move your email to Office 365 and keep your current public-facing website.
Thanks,
Allen Sun