Vicki, is this your first trip to Europe? You don't appear to be accounting for how wiped out you'll probably be after the overnight flight (which means you'll likely have only two really useful sightseeing days in Paris) and the hassle of changing hotels so many times. In the amount of time you have, I'd shoot for Paris and just one other area.
Nice is a nice city, but most people who go down to the Riviera want to see some of the small towns as well. By the time you get there and get settled in, you'd only have about 2-1/2 days. Although that's doable, you'll miss a lot more than you see, and you'll spend a lot of time in transit.
There's a lot more to see in Normandy than just Bayeux, and Lyon is a large city with a lot of sights of its own.
With the caveat that I think your target list is too long, the only sane order for tackling it is north to south. With this extraordinarily packed destination list, I'd want to get to Bayeux on my arrival day (making some use of that jetlagged period when you probably won't accomplish much in Paris, anyway). So I'd go Bayeux-Paris-Lyon-Nice. The difficulty is in knowing what to do about the train ticket to Bayeux. You may be able to snag a very reasonable fare if you buy far in advance, but saving a lot of money means buying a non-changeable/non-refundable train ticket. It's impossible to know when you'll clear the airport upon arrival and be able to start your train journey--even if your flight is on time. The usual recommendation is to wait to buy arrival-day train tickets until you are on the ground at your destination, but that will make the trip to Bayeux rather expensive.
It is not recommended to drive on your arrival day due to the probable sleep-deprivation and jetlag. For your itinerary, trains will be much faster than driving.
What time of year is your trip?