Hi and welcome to the forum!

1. Do you know yet where you'll be living? I only ask because you will find that houses in France are generally much smaller than those in the US, and the rooms are generally quite a bit smaller than what you're used to in the US. IKEA had to redesign its furniture for the US market because it was too small. So, the same stuff bought from IKEA in Europe will actually be smaller than what you bought in the US.

If you know where you're going to live and the furniture will fit, go ahead and take it. It's really nice to have "familiar stuff" when you first get overseas. And IKEA stuff is usually easy to move, since you can take it all apart for easy shipment. But you should also consider that, if you replace your furniture (with more IKEA) when you get to France, it will be smaller and possible a better fit with where you are living.

2. I would advise against shipping a car. Although Corollas exist in Europe, they aren't the same as the Corollas you get in the US. Also, there is a very strict inspection system in France - to get the initial registration in France, you take the car to a special government inspection site to make sure it is up to French standards. Normally, you also have to pay VAT (19.6%) on the car if you haven't paid it before, though they may not assess that if you can show that you paid sales tax when you bought it. (I'm not sure if they consider sales tax to be the equivalent of VAT.) It's complicated - and probably you can do better to get a car that is already registered in France.

3. I've moved from the US to Europe twice. Unless you're trying to save money any way you can, I'd advise using an international mover and getting adequate insurance for your stuff. The big hassle is clearing your shipment through customs when it arrives, and that's one of the services the mover should be able to provide. The list you provide for insurance purposes also serves to clear your shipment through customs.

When you move to France, you're entitled to bring in all your stuff duty free, though you'll be charged VAT (19.6%) on any items you have owned for less than one year. Having a customs agent to clear your stuff makes life considerably easier.

All this is, of course, my opinion and you'll find folks with other points of view.
Cheers,
Bev