The certificate is actually called a Vaccination Certificate and any certified vet can do it for you. I have a copy of one that most vets use and I can email it to you if you like.
As Stray Dog said, you are more anxious than your pet. IMO think pets are better off in the pressurized luggage compartment than stuck in a tiny little crate that they can hardly turn around in, under the seat. If you have to put your pet in the pressurized luggage compartment, you'll probably buy a bigger crate which will be more comfortable. All the pet will hear for the entire flight is the hum of the engine and that will probably just lull your pet to sleep
No, it doesn't seem quite right. They usually send a confirmation email when you book a pet. I would request some sort of confirmation via email.
If you have any questions feel free to email me at Natasha@BARK-Taiwan.org