Best way to develop a new theme on a live site, with new content? [closed]

You should never do work on a live site (of any size).

Copy the existing site to another server, give it a temporary domain and block search engines in some way (robots.txt, http auth, …). Depending on the work you do, you could stay connected to the same MySQL server, but it is usually recommended to also copy the tables to another db.

Now you have an environment where you can do all your work without any fear that if anything breaks, the live site will suffer. Once you are finished, simply upload your themes and plugins to the live site, install them, et voilà.

You can even use the regular theme update method for custom themes not hosted in the official repository.