I would recommend that you save Aix for another trip, you won't do Provence justice in such a short stay. Just stay the entire time on the Cote. You won't be bored!
As far as a car on the Cote, you do not need one in Nice, but you will need one to be productive for most of the other areas. YOu just won't see the best things on a bus or a train, and after reaching your destination you will be stuck without a taxi. The Arrière-Pays, the back country, is full of wonderfully hilly roads, switchbacks, and gorgeous vistas winding up in fantastic hilltop towns. One, La Turbie, has 2000-year old Roman ruins and a fantastic restaurant, Hostellerie Jérome. The other don't miss restaurant on the Cote is Mirazur, down on the coast in Menton. Some other don't miss hill towns are St Agnès, Gourdon, and Peillon.
Parking is possible just about everywhere but Nice, just be patient, and use the lots in the bigger towns.
Do you want a produce market, or a market with clothing and crafts? For food, Nice Cours Saleya market is every day but Monday, and is the largest in the department. Clothing and crafts? That's actually the Friday market in Ventimiglia, Italy, just over the border, with 400 vendors!
These 2 restaurants, in one word, are SPECIAL.
There is so much to do on the Riviera for 5 days. Time will go fast. Beaches, markets, historic homes, great museums, and food!
I can be more specific and detailed if you have a particular interest.