Major airlines do not guarantee the safety of any living thing in the hold of the aircraft.
Unless something has changed, my understanding is that small animals (cats, dogs under something like 10 lbs) are allowed to fly with you in an approved carrier.
Any other creature must go in the belly of the plane with the baggage. You will be charged the price of a seat, and in return the airline does not guarantee any particular temperature range for the baggage hold nor even that it will remain pressurized. This is for domestic flights in America - I cannot speak for other air travel situations.
As a proud owner of a 60lb dog, I have sometimes been in the position of having to arrange alternate transportation (usually driving a car) or altering plans to accommodate my pooch. I simply cannot take the chance that something were to happen similar to this story. It is tragic, no doubt, but the provisions airlines make are very clear and most of this risk could've been avoided.
Edit: I've thought about this issue a lot. I think it would be in everyone's mutual interest for airlines to offer a safe way to transport pets or other small animals. Assuming that's a non-starter with existing carriers, some other business could serve this niche effectively. I suspect it could be done profitably.