Here is a thought. Do consider staying in a hotel if this is your first time in Paris (or even your second)! The hotel staff are a very valuable resource for all things Paris: metro, good value restaurants, directions, and all of the things that the Parisians know but you don't! The hotel staff give you peace of mind and can save you hours of time or several bad meals, or give info on the frequent strikes (like almost all of the museums in Paris now), etc. 7 nights is not that long in Paris and you don't want to waste time and money.
Also are you really going to cook in the apartment? If you think you will, check to see if there are any spices and basic foods provided. By the time you purchase these for a couple of meals, you might as well have eaten out!
Hotel cost like apartments are often based on the distance from the prime tourist locations. Also make sure that you are near a metro stop (no more than 5 minutes). I am a travel consultant and France Specialist who works with clients on moderate budgets. They are always happy with my suggestions in this area.
I also recommend Rick Steves France and Paris Travel Guides because he provides the best information on the metro, museums, way of life, fun things to do, etc. Check out his recommendations for hotels altho his preferred area is not close to any tourist area except the Eiffel Tower.
Good luck on your trip! Just dress warm since it can really be chilly walking around in Paris in the winter.
Also, what is your budget per night for a Paris hotel? Remember that it includes all taxes and often continental breakfast so add that on to your budget.