Anecdotal experience: My daughter broke her arm, was in a cast, and was given an immediate Fit to Fly determination. We are on long-term Visas, I simply asked the question if it was the wisest choice to fly within a few days due to upcoming travel back home.
I asked if I should delay my flight (my initial preference) thoughts of swelling at altitude in a cast, a child...
I was cleared to fly and then provided the certificate.
Leaving France, you will need an inspection by a veterinarian and then the French Government will provide the export papers. I did have difficulty with the Government office in Paris. Had to go back to the veterinarian and back to the Government office. Plan a few days in advance in case something is needed.
Vaccines must be kept up to date, microchip and a Rabies Titer test to leave Thailand and return to the EU. Be mindful of the dates. The titer test must be done after 90 days from last rabies vaccine and 90 days before travel. (I'm fairly certain of those dates, but please check)
Timing of the titer/vaccine did result in me cancelling flight plans.
I recommend using Relo4Paws for export and import permit to and from Thailand. I could do it myself, I prefer to use them as requirements do change and I don't risk making a mistake.