May-June will give you more daylight for sightseeing. The weather should be pretty good. If you choose this time frame, get started soon on the apartment search because the best places will be filling early in the year. September is a busy and expensive month in Paris because of trade shows, but you wouldn't go wrong with the fall months either. In that case, start looking for an apartment in the spring.
Your plan makes a lot of sense to me, you're lucky to have so much time. Two weeks in Paris won't be too long. You'll get lots of advice here about Paris apartments, including cautions about current efforts to enforce local laws against short-term rentals where taxes aren't paid. Your safest bet would be to use an agency that itself owns the properties, rents them out, and pays taxes on the rental income. Or try to get an assurance from the individual owner that taxes are being paid.
Except for Normandy, I don't know "the coast." France really has three -- English Channel, Atlantic, and Mediterranean -- and you have time to drive from near Paris into Normandy, Brittany, and then down the Atlantic coast to St-Jean. You could also spend some time in the Loire Valley looking at chateaux and such, and/or the Dordogne further south. Hopefully you can do this non-Paris part of the trip by rental car, getting it near Paris (a short train ride, say to Rouen or Caen) and dropping it at St-Jean or wherever you want to finish. If you continue into Spain you'll incur a big drop-off fee unless you return the car to France. Consider flying home from Bordeaux or Toulouse to save the time and cost of backtracking to Paris by land -- unless of course you want to see more inland sights.
Good luck planning this trip. You'll get lots of help here with focused questions.