Am I eligible to be an auto-entrepreneur for only nine weeks per year?

I would be most interested in your registration service for auto-entrepreneur. I am a self employed UK ski instructor working annually in the French Alps (December to end of April). I have the E101/A1 document entitling me to work in France but pay my tax contributions in the UK. My marketing, payments and bookings take place in the UK throughout the year. My winter season is split into two sections.

Firstly I am a UK sole trader, tour operator running intensive ski courses. I organise the entire package holiday for these clients and I teach them during their stay. These packages are largely for bookings in January and March. Secondly, over the Christmas holidays, February and during the Easter holidays I teach regular UK holidaymakers.

Other information:

• I rent an apartment all year, but spend six to seven months in the UK.
• I have a bank account with the La Banque Postale, but not a dedicated business one.
• I’m hoping to get this in motion before the EU referendum in case it may help me if changes happen with working in the EU etc.

1. I would like to establish this as my French auto-entrepreneur business (only trade for nine week per year) - is this possible?
2. Do I need a document of residence?
3. I don’t pay the bills in my apartment, except wifi with SFR - is that sufficient?