The only thing you get out of them if they are later than the guaranteed time (which sometimes is 12:30 or even 5:30 more and more on Next Day Air shipments) is the price you paid for the shipping. They won't guarantee or insure anything live. If you tell them you want insurance, they'll tell you to ship with someone else. If something is late and you mention to them that it's a live animal, they seem to care a little bit more and try to get it figured out, but it still doesn't really change anything as far as their liability.
Technically, to ship live animals with them, there's a paperwork process you go through, and they basically approve of your shipping methods and ask what you're shipping. So if something is missing or late, and I do call and tell them it's an animal, they'll usually throw a little bit of a fit and say they don't ship animals, but I just tell them to go look at my paperwork and to find my damn animal. Even their own operators don't always know the rules until they can find proof that you have it done. If you don't have the proper paperwork filled out, they might make a bigger deal about it. Also, they don't ship snakes, amphibians, and a few other things, but lizards kinda slip right through the cracks of their paperwork. I have also been shipping quite a few tortoises with them lately and that's apparently ok also.