Do we need a car to see Lyon and surrounding Areas?

I lived in Lyon for several years. Generally, in the city, a car is a liability more than an asset unless you need it to transport heavy or bulky items. Best to not have a car when you're staying in Lyon. The transit system's métro, trams, buses, and funiculars are far more useful than a car.

But for trips outside of Lyon, a car would make things easier and increase your flexibility. So, consider grouping your Lyon activities to one part of your agenda and the outside Lyon activities to another part of your agenda and arrange the rental car for those times when it would be most helpful rather than the times when it would be a pain in the a**.

If you think you might do a day trip from Lyon, then I would say yes to the rental car. In that case I would suggest Roanne as the town for this day trip, ie northwest of Lyon. Went there a couple of times in the late 1990s, interesting historically I found as respects to Napoleonic history. I am glad I saw this town.

Still, if that sort of history is not within your intertest, the town is worth visiting and exploring.

A number of fun day trips from Lyon are best done with a car - I'm thinking of Pérouges, for example. Annecy is a lot faster with a car. To get to vineyards and cute villages in the Beaujolais, yes you need a car. And not too far from the Beaujolais, the château de Fléchères, a personal favorite, is really not accessible by public transportation.
As others have said, I'd really discourage you from driving in Lyon. There are a lot of one-way streets, and poor signage on some major routes can make it challenging even just to get out of the city. Definitely need a navigator as well as a driver! But it's a wonderful place.

I'd say you have a good plan, but be aware that Avignon really doesn't support entry by automobiles. You'll need to park outside the city and enter on foot.