I am quite surprised that you found the link I sent you useless. It has links to French government websites with actual information about resident permits.

If you work for a public international organisation you do not pay income tax in the country in which you live, regardless of whether it is Switzerland or France.

But by choosing France you lose some of the privileges that you get if you live in Switzerland -- spouse employment on a Ci permit, for example.

What other actual information do you need? Yes, they may stop you at the border and ask to see your visa or your residence permit. Or they may not. The more time you cross it, statistically your chances of being stopped increase.

I would say that you need to sort out your work-related issues first -- will surely need a copy of your contract when renting a flat wherever it is. Then go and talk to real estate agencies that deal with rentals on the French side -- there are dozens if you go down the main street in Ferney-Voltaire, and I am sure they can enlighten you further.