Each carrier has a whole slew of regulations and policies. You should be able to find them on their website. Or phone the booking number.
The key is the regulations of the originating, transit and destination countries. Check with the relevant sections BEFORE making any bookings. Quarantine will apply, the only question is how long.
Animals are boarded in special kennels, placed in specific parts of the hold with heating and checked at interim ports of call. The flight crew will know that they have animals on board.
This is routine and happens every day.