Paris for 5 days and renting a car

It's best to stay in Paris while visiting Paris.
Re the rental car trip in Normandy, my experience has been that It might be best to take the train to one of the larger Normandy cities/towns and rent a car there (probably at the train station).
If instead you choose to rent the car in Paris, you probably will want to rent at the CDG airport or another location near the Paris outskirts which will avoid having to drive in Paris.
I've found driving in Paris, even for a relatively short time, does not enhance your trip experience. We'll see what others say about that.

If you want to see the sights of Paris, then staying IN Paris is, of course preferable, since you won't be wasting valuable vacation time travelling to and from your accommodations each day. Find a place close to a metro station. With only a few days notice, you might not be able to be too picky about your accommodations. If looking at Airbnbs, be sure they are legal and list the 13 digit registry number.

I'd recommend taking the train out of Paris and renting your car in Normandy. If you're headed to the D DAY beaches area, you'll find rental agencies very close to the station in Caen.

Where can you find the 13 digit registry number on the Airbnb site?

Under the description it will say "more information" or "read more about the space", click on that and towards the bottom you'll see it, if they have one.
See this listing:
https://www.airbnb.com/rooms/534650?location=Paris%2C%20France

If you have never been to Paris before, consider a hotel rather than an apartment rental. The front desk of an hotel can be very helpful and apartments can have a lot more unknowns. Do not stay outside of Paris if your objective is Paris.

If you're going to Normandy after your stay in Paris, rent car from an office near the southwest part of the city. Specifically near Porte d'Auteuil from where A13 departs Paris for Normandy or from Porte Maillot from where A14 departs Paris to reach A13 farther to the west. Absolutely avoid the airports, CDG or ORY as they are on the eastern half of Paris meaning you'll need to negotiate a lot more traffic to reach Normandy than necessary.

Also, booking.com has apartment rentals & they list the 13 digits towards the bottom of the page in The Fine Print section, "License Number:"

Paris is a very large city if you want to see Paris ; you should stay in Paris.
I would make use of the high speed train and head out of the city faster that way and with less stress than a car.
Rent a car after the train ride.

A train from Paris into Normandy would possibly be the easiest, picking up a rental car at the train station.

Just a thought...not sure what day you’re leaving Paris? But before finalizing your itinerary, check the hours of the local car rental agencies for the day you’re thinking of traveling. Some close for long lunches, then close for the day at 5 or 6 pm; and most if not all are closed on Sunday.

Have a wonderful trip!