Depending on the address where you plan on staying, the 7eme arrondissement is extremely quiet at night, and during the day, there can be long, long stretches where there won't be any cafes if anyone needs a toilet or anything to eat.
On the other hand, there are loads of little parks all over Paris, many with free playground equipment for kids to use. So, you don't have to limit yourself to the 7eme or in the areas where there will be just one or two major monuments. You can get anywhere you want to go in about 20 mins by Metro.
As someone mentioned, I wouldn't recommend using AirBnB, and it's becoming difficult to find legal apartments since the new laws were passed this summer. It is possible that you might risk a last-minute cancellation unless you are certain that the rental is legal - then you'd have to scramble to find another place to stay.
If you want the best and biggest market in Paris, you might think about staying in the Marais, near Bastille. Market days are Thursday and Sunday from around 9:30 to 12:30 PM. There is a playground at the end of the market opposite Bastille, as well as one at Place des Vosges, a short walk away. Both are open every day.
There are legal apart'hotels in this area - what they might lack in charm is redeemed by having clean sheets and towels, a kitchen, laundry facilities, real beds and a little more space.
CitadinesBastilleMarais would be the closest to the Bastille market.
AdagioParisBastille is another legal apart'hotel which is quite near the Marche Aligre (open every day except Monday) and in a much less tourist-oriented neighborhood.
Both of these apart'hotels are convenient for transportation via Metro and Bus.