Paris & Lyon in February

I’m hoping to hear from some of you who have experience, especially in Lyon.

Best walking trails in Lyon would be along the banks of the Rhône and in adjoining parks. You can go from Parc de la Feyssine to Parc de la Tête d’Or then along the river to the Parc de Gerland. Parts of Parc de la Feyssine are quite wild and there are far fewer people there than Parc de la Tête d’Or which has nice gardens but a bit desolate in February. Depending on where you end up staying, I’m a fan of the Voie de la Dombes in Caluire/Cuire - nice wide smooth trail, but not as convenient to center city.
As for nearby villages, it depends what you’re looking for. Pérouges is old and attractive but gor me more of a day trip than somewhere I’d spend a lot of time. If you want countryside, the Monts du Lyonnais to the west of the city are nice, several attractive villages (Bois d’Oingt, for example) but again, not much by way of major sights. A little north there’s the Beaujolais area where obviously you can find wine. And Villefranche-sur-Saône, Macon, …. Crémieux to the southeast of Lyon, not far from the airport, has some charming old areas.
Basically, there’s a lot of possibilities so you need to decide how far you want to travel, what you’re looking for, etc. Essentially all the places I’ve mentioned require a car.

Without a nightly budget, we're just guessing, but you might have a look at the aparthotels in Paris, such as Citadines. They're all over the city.