"can I get American prescriptions from an American doctor filled in France?"
Not directly. They would need to be re-written by a French doctor, unless they are sold in France without a prescription. Which drugs need prescriptions, and which are available over the counter without a prescription, varies by country; it is not uniform across Europe.
The number one thing to make sure you don't lose your prescriptions is never to put them in checked luggage. Make sure you have them in carry-ons. If there's a possibility that you may have to check your carry-ons at some point (say, if one of your flights is on a very small plane, which just happened to me on the Madrid to Biarritz flight), have a separate bag that you can pull out with your meds.
Next, set up a system to make sure you don't leave a hotel or other accommodation without them. Set a reminder in your phone, or put the meds inside your shoes, or only take out what you need for that particular stay and leave the rest packed in your suitcase - there's lots of ways, but make sure you implement one, and use the same method at every stop so you don't forget or get it mixed up.
Depending on the severity of your medical condition, you may want to research English-speaking doctors in places you will be going.