Given that it's the month of August, many people would recoil in horror at the thought of the Cinque Terre but it is reachable for you in your 7 day trip - and if you like hiking, with a decent view, you'll get that at least. Plus a million other tourists, as well.

Lots of day trips if you base yourself in the Nice area, although you might be better advised to use public transport for some trips and avoid the very busy roads.
The train brings you to Cap d'Ail from Nice where you can walk all the way to Monaco on the coastal path
Beyond Monaco, you can ascend to the old town of Roquebrune or walk around Cap Roquebrune by a coastal pathway (using Roquebrune rail station as a start point).

Villefranche station gives you access to the Cap Ferrat coastal path, a pleasant 2/3 hour hike.

Your car may be of more use to get to hill towns like Vence, St Paul and Eze although parking might be a problem in the busy season.

Toscane is a five-hour drive from Nice according to Google Maps. Might be longer in August. I'd stay in France. Lots to do around Nice.

A different excursion from Nice is the little Train des Pignes back into the hills north of Nice. Plenty of hiking opportunities along its route. Most people doing a day trip use Entrevaux as a turn-around point.

For August, I'd make hotel reservations ahead of time.

SeeDee, Mimar,

Thanks a lot for the recos!
So far, taking both trains to move around Nice seems a very good option; I'm aware of the amount of tourists we will be and having the right choice of transport will be key.

Driving for 5 hours to Toscane doesn't sound bad, especially because of the great views we can get in the road, but I guess that the main driver of the decision is to spend enough time in every place to enjoy...

If we stay on the French side; would you recommend to spend all the 7 nights in Nice and do day trips, or should we split btw Nice, St.-Tropez and Cannes 3/2/2 ?

You realize that August is vacation month in Europe - the seacoast will be very crowded. That being said, Nice is a wonderful city, with lots to do and see, and a reasonably nice beach. However, you might want to split up your week between Nice and a smaller seaside town, such as St. Tropez. Another very nice little town just outside St Tropez is Sainte Maxime. Not quite as 'jet set' vibe as St. Tropez, but very pleasant, with some great beaches and bike trials (you can bike to St. Tropez). I would do 4 nights in Nice, 3 in Sainte Maxime or St Trop. The Grand Cornice or golden road is between Ste Maxime and Cannes. Because of the season, make your reservations early.