@Patrice94, I feel your pain! The 1042 is a nightmare. In my other business, I'm supposed to file it when I pay foreign clients. Since that doesn't happen very often and their income is minimal, I just pretend to forget all about it. 🙂
However, since all your US income comes from that one source, you're probably going to have to get it straightened out. I would definitely get a US CPA on board.
Where is the house? Is it in a state where Airbnb collects occupancy taxes on your behalf?
The US and France have a treaty for relief from double taxation. I only know how it goes from your country to ours: When my clients earn money in France, we have to file a form 6166 with the IRS. This proves to the French tax authority that the US client is paying taxes in the US on income earned in France. This then reduces the amount the French can tax. I'm guessing there's a similar form you can file with the French tax authority that will tell the IRS that you are paying French tax on income earned in the US.
Do you have a French CPA who understands all of this?