huguesthomas

Yes, you would, and you might find Paris wants proof of French residency, not just citizenship. Most consulates don't like you 'consulate shopping' and will only accept cases from those who are actually resident there - no idea if Paris is among them though, so you can always give it a try. 
 
Good luck. 

No, you need to contact Paris directly - https://travel.state.gov/content/travel/en/us-visas/immigrate/national-visa-center/immigrant-visas-processing-general-faqs.html#ivp13

If your case status says at NVC, you need to contact NVC to transfer your case through their public enquiry form. Include your proof of citizenship in the attachments. That is what has worked for the multiple people I spoke with.
 
Switching consulates is pretty common and straightforward. Just make sure you know are prepared with all your documents and how to go about medical/biometrics for the consulate in France since you'll have to leave the country for a few weeks to go through the process.
 
Here is all the information for the green card consular processing at the consulate in Paris - https://travel.state.gov/content/travel/en/us-visas/Supplements/Supplements_by_Post/PRS-Paris.html

Yes you would need to go to France for the interview. You can request to pick up the passport at the Consulate. So within a couple days. But generally most Consulate give 6 weeks notice of interview date.