A much better plan Gordon.
Definitely go with at least 4 for Paris, it is a large city and the attractions are spread out. Even 4 is not enough to see everything so you have to prioritize your interests.
If you want to do a day trip to say Versillies OR Champagne Region, etc... you would need to make it 5 nights.
I would recommend you think about a day trip and go with 5 nights Paris, and the 7 or 8 you have remaining in the Amalfi Coast with a 2 night side trip to Capri sounds like a perfect trip.
The city is broken out into different Arrondissements so picking one to stay in is your first course of action.
You have to compromise on something as any one location will mean a good walk or public transit to certain things.
I think if you like cobblestone streets and lively markets, tons of dining and shopping options the 5th or 6th is a perfect spot.
I kind of lump those 2 together since there is no clear division of where it changes to my knowledge at least near to the river they have the same feel.
the 1st is good for being centrally located, scenic, quiet but is not that lively and lacks restaurant and shopping selections ; this would be my second choice.
Your really won't find the large hotels and chain hotels in the areas that I am referencing.
I first assumed I wanted to be as close as possible to the Eiffel Tower, but the more I looked into it and what I wanted to do and experience the area near the Eiffel Tower is not as appealing to me. It is more spread out there, more modern, more car traffic, etc... But if you really want to look at the Eiffel Tower from your room each night, then maybe you compromise at the expense of the benefits of staying elsewhere, as I mentioned it is a compromise no matter where you stay.
Since you mentioned you have been to Rome.
I guess a similar comparison to where the Eiffel Tower is similar to where the Colleseum is in Rome vs. the 5th is more like where the Pantheon is. Just double or even triple the distance between to get an idea on size.
Paris is much more of a working city, with tons of traffic ; more like NYC in ways than Rome in that regard.
It has amazing sites and history for tourists but is not a tourist first city.
If you really like luxury shopping or want to be next to a specific place there may be better options for you, just giving you the general idea that sites are spread out.
You may want to do a boat/bus combined hop/on hop/off the first day.
Especially if you arrive in the morning from a flight it is a good way to take it slow, get over the jet lag and kind of get the lay of the land.
You should book your time/ticket to the Eiffel Tower and Museums in advance to avoid lines so figuring out which day you want to see what is a good plan.
Keep your daily plans open and not packed, the heat and lines in August will make you regret a packed itinerary.
I have been to both the Amalfi Coast and Cinque Terre and though I like both places a good deal, I think there is not much to gain by including them both on a single trip. The August crowds in CT could make it less enjoyable ; they could do the same in Amalfi but that is a larger area so more room to spread out, also much more to see and do there.
Amalfi Coast is more Wow scenery than the CT. CT can be in places more picturesque, the AC more grand, blow your mind kind of views and of the 2 most definitely my preference for a long stay especially.