I work in dispatch I'm usually involved in the process of making sure everyone who needs to be notified has been notified about live animals.
ALL flights that carry cargo must have cargo documentation on board which the crew will usually read through so they have an idea of what's downstairs. Any 'special' cargo (such as perishable goods, livestock, dangerous goods etc) will be specifically highlighted for the crews attention. The dispatcher will also ensure the crew are aware of any animals onboard. In addition to this, either the capt or first officer will usually physically see the animal prior to it being loaded on the aircraft during their walk around. Where I work at least, any animals on board are usually put in the SI section of the loadsheet too for the flight deck to see.
As CEJM said, I'd also highly recommend strapping a bottle of water to the outside of the crate so the ramp team can top up the dogs water prior to departure, and also post arrival. I know that myself and many of my colleagues tend to get bottles of water off the crew on inbound flights if there is animals on board to top up their water.
Hope this put your mind to rest a little bit
