..how many days in Rome and then the CT?
There's no good answer for this unless we know what sort of time you have and what sort of things you are interested in. I've spent nearly 2 weeks in Rome (over 3 trips) without seeing more than a fraction of it, and 3 nights in the CT without more than scratching the surface of the available hiking trails.
Outside of your question about the CT to Paris, what do you want to do and see in Rome and the CT, and how much time do you have to do that?
Editing to add: while the CT is often mobbed during high and shoulder seasons, if you are a hiker then you can lose a big chunk of the masses by heading up into the hills during peak visiting hours. Even just higher points of some of the villages themselves will experience less of the crush, as many day-trippers are unwilling to make the necessary steep climbs to upper churches, cemeteries (great views from those!!!) etc.
Sorry but I'll disagree with, "their destruction as a tourist destination largely due to over publicizing by Rick Steves." RS guidebooks are not widely used or his shows watched outside of North America. Visitors to the CT come from all over the world; we hiked with more fellow trekkers who didn't have English as their mother tongue than those who did.