Which category of self employment and health care access in France?

We are a family of three - husband, wife and daughter aged six years. We have owned our house in France since 2002. We moved permanently to France at the end of July. For the last 20 years I have worked in various countries. Each time I took a new contract the employer and I paid National Insurance (NI) in the UK for the first year. After that it was not possible to continue paying full NI contributions. I have been non-resident for the last five years and have not paid income tax. Since May 2003 my family have lived with me overseas and the employer provided private health insurance for all as part of my contract.

Now we have permanently settled in France. I plan to continue to work short missions while my family stay in France. I hope to work for about eight months of the year abroad and spend the rest of the time at home. The yearly income would be about 50,000 euros. The money will be paid from Switzerland in to our French bank.

CPAM said that to register for the state health system we need a S1 form from the UK. The NI refused to send one as I have only paid NI in UK for one out of the last three years. The NI will send a legislation letter stating that we are not entitled to use the NHS. The staff suggested we either take comprehensive private health insurance in France or register as self employed asap. Is this correct?

Which self employment (start a business in France) category should I apply to join? Is it likely that all this could be sorted out giving us access to the state health system by the end of October?