IMO, no, it's not really doable. At least not in an enjoyable way. I see you disregarded the suggestions in a previous thread to just stay in London and fly to Italy.
1 1/2 days ( the first in a jet lagged stupor) to see London on your "first and only" trip? That's really a crying shame. It deserves Soooo much more than that.
Fly to Paris? Why? It's likely no faster ( accounting for time required to get to and from the airports) and certainly more stressful than taking the Eurostar from city center to city center. Or if you're determined to fly, fly into Orly, which us at least on the south side of the city.
All the usual car rentals are available in Paris, including a lot of American brands. There is no best. Check out Autoeurope. They're a well known 3rd party company that will show prices etc from different rental companies. Be sure to check the open hours for your pick up and drop off locations. They may be closed on Sundays and for lunch. Rentals in Europe come with the same liability insurance as in the US, but you need to consider collision damage insurance. You may have this with one of your credit cards, or you may need to add it onto your rental agreement.
Get a manual transmission due to parking?? That makes no sense. Compact cars with manual transmissions are the cheaper option in Europe. A bigger car and/or automatic transmission will just cost more, and they're in shorter supply.
A map is a map is a map, whether it's paper, a GPS, or a download from Google maps. Driving through Paris can still be, if not a nightmare, then at least stressful. Especially if you've never driven in Europe before. Before leaving home, study up on French driving laws and traffic signs. And don't forget to buy an International Drivers Permit for each driver from your local AAA office.
Only one day to see the DDAY beaches. Don't try to do this on your own unless one of you is a military historian. Sign up for a tour like those given by Overlord tours. And you had better pray that that one day isn't pouring down rain.
Flying out of Paris on the same day you drive back from Bayeux. It's a MINIMUM 3 hour drive in ideal conditions (car doesn't get a flat, good weather, no accidents blocking traffic, no construction or other impediments, not getting lost trying to find your drop off point). Unless you start out really early in the morning and have a late afternoon flight, that's taking a risk. Up to you.
As you can tell, this is not a trip I'd undertake given your time constraints. I agree with the posters on your other thread that if you really want to see the Normandy beaches, do it at the end of your cruise when you go back to Paris. And hope for decent weather.
