I have no input on Nice per se, or using Nice as a home base. I'm sure there's good stuff to see there.
Yes, absolutely, by all means, extend by a day. More if you can. If you can't, then work with what you've got, and accept the limitations that come with it.
Why the obsession with how many days? For most people, I think that their air fare is the single most costly item for a trip to Europe. Given that (and assuming you don't have lots of "extra money" and you don't have the luxury of going to Europe very frequently), it just makes sense to get the most from the money you spend on air fare. That means staying long enough to get more out of your trip. Personally, I don't think it's worth spending all that money to get to Europe and back unless I'm there for (not counting travel days) a minimum of 2 full weeks, but everyone's different. And of course, everyone has their own constraints.
we would actually be in Europe from Saturday morning (the 6th) and flying out Saturday (the 13th) morning (the latest)...
So Saturday the 6th is your arrival day. That's a travel day, and I wouldn't count that as really part of your trip (by my reasoning, your trip actually starts the next morning when you wake up).
On your arrival day, you will be jet lagged and wiped out (most people can't really get any meaningful sleep on a flight, and many of us start off stressed and short on sleep from pre-trip tasks, making it even worse - try to begin well-rested and do what you can to catch catnaps on the plane if you can). On your arrival day, on a too-short trip, you should press on to your actual destination (Nice, or wherever) - by train or short-hop low cost flight (no driving upon arrival, it's dangerous). Saturday the 6th is pretty much a write-off. Get to Nice (or wherever), walk around and get some fresh air, your primary task is to stay awake until after local dinner time, then crash and burn. Wake up the next day and your trip starts.
You fly back on the 13th - from Paris. So you have 6 full days in Europe.
But there's one more fly in the ointment: you fly out of Paris. If you miss your flight home from Paris, you are in trouble: you would probably have to buy a very expensive ticket for a new flight. Because of that risk, many of us plan to be in our departure city the night before we fly home. If you're basing in Nice (or anywhere distant from your departure airport), keep in mind that lots of things could prevent you from making it back to Paris on time. A rail strike, for example, or bad weather, other things. So, you have 6 full days in Europe, but the last day of your trip should probably end in/near Paris (to avoid exposing yourselves to the risk of missing your flight). Some people don't worry about that, and plan some short connecting time to their flight (and they usually get away with it). But the potential cost of missing your flight is worth considering.
Having more days to spend there mitigates all these worries, having a very short time there (which I think describes your trip) makes things more risky/expensive/constrained. Not saying you can't/shouldn't pull it off, just saying you should be aware of these things going in.
Hope some of that helps.