I"m sure someone will be along shortly to explain in detail what you have to do.
But I know that one thing you have to do is register with your mairie as a tourist accommodation provider.
As for whether you need to register as a business, the rule used to be that if it’s your main source of income, or possibly a significant source of income, you do have to register. You will then pay full social security contributions based on your income, because this will be your route to healthcare. If you have another source of income (and therefore you have healthcare via that other activity) and the B&B is just pocket money, you’re not necessarily obliged to register.
I don’t know if this is still how it works, and I can’t remember what percentage of income the B&B can represent before you have to register, and there might be an earnings ceiling above which you are obliged to register as well.
But I think that’s roughly the idea.