Living in France doing Telecanvassing for UK Company

I have been approached by a UK company to do telecanvassing for their sales team who are based in the UK. Making appointments basically for the sales team. I would be invoicing the UK company for an agreed sum monthly and being paid in Sterling into my UK bank account. I have lived here for 8 years now and not worked until now due to ill health. I am not going to be employed by them, just supplying a service for them and invoicing them for that service monthly.

1. Can I register as an auto entrepreneur and declare my income & pay tax & health care charges here in France or would I have to pay in the UK also?
2. Also the UK company need to know how it affects them - If I work purely for them and nobody else, they seem to think that they would be liable to pay Employers PAYE etc in the UK?
3. They have said that my invoices need to be non consecutive numbers so that it shows that they are not the only ones employing me effectively? I am very confused? I have asked the UK TAX office and they are going to come back to me.
4. My husband is also thinking of starting a fishing guide service here in France. Would it be better for him to set up also on the auto-entrepreneur system or just put that under the same umbrella with me thus getting around the consecutive invoice number system?