, I don't know about other cats, but mine never pooped or peed during flights (we had 4 long distance flights, we used soft carrier). Cats tend to keep their space clean and feel safer when their owner is nearby. I guess it influences what they do or not do during the flight.
And it is easier to get the approval for Pet in Cabin rather that Cargo or as Handled bag. Your cat would be better next to you during the flight. You'll feel better too. I flew with my cat between continents 4 times in the past year, and I'm glad I chose taking her as Pet in cabin. It makes things much easier.
You don't need a long term visa to bring your cat in the country, 100% sure. I brought my cat as well a few months ago while I was on tourist visa. Importing your cat and your visa are not related to each other. Theoretically you can import the cat and leave it in Thailand in other hands even if you return to your home country, as long as it was belong to you for a decent time before importing the pet.
Moreover, you don't really need a company to take care of the export/Import process. It is straight forward and you just need to send a few papers in advance to Thai govermental vet authorities and they'll send their approval. The pet export/Import company doesn't make things shorter (just get paid for doing what you can do yourself easily, you anyway have to submit the same paperwork). PM if you need more direction.
The hospital can take care of her visa extension directly with the immigration. Every international provides this service. Please ask the hospital ASAP (usually the Customer Support counter).
Adding my best recommendations on Indus on Sukhumvit 26. Great food, service in a very nice setting. Reasonabley priced. They keep the high standard for many years.