If you're looking for apartments in Paris, you'll have to decide what kind of sleeping arrangements you can tolerate, as well as how many full bathrooms will be included. Expect the apartment to be smaller than you would like, and it might not have an elevator and will not have air-conditioning.
The price range for a family of 6 would mean a deluxe apartment, and it depends on how far in advance you could book one, because they are normally booked up a year in advance.
If you want 2 or 3 separate bedrooms, each with "real beds" and a full bathroom, the price range will be around 2000 to 2500 EU per week - possibly more. If you are staying less than a week, prices will be higher.
Damage deposits and cleaning charges might be included or listed separately.
If staying less than a week, you might try finding adjoining rooms in an apart'hotel, such as Citadines or Adagio.
Le Roi de Sicile and Hotel Residence Henri IV are also nice apart'hotels.
All of these locations have very good transportation options.
AirBnB and vacation rentals in general will vary. You need to check the actual address using GoogleMaps Street View to be certain that you're comfortable in the neighborhood.
No idea what the other situations would be in the cities you mentioned, but I'd expect them to be even more expensive than Paris is.