Which animal is most likely to break free or escape (either due to strength or intelligence)?
Which animal is most likely to go absolutely crazy and become a danger (to himself or others)?
Animals that get extremely bored or stressed out can self mutilate. Psittacines (parrots, macaws, etc) are notorious for this.
Are they any specific diseases or illnesses that you worry about as a zoo that can cause a big problem for the animals?
Tons. Too many to list really. The most common first sign of trouble is a change in diet consumption. Animals mask illness so that they do not appear weak. Some do it so well (especially birds) that by the time you see any real symptoms it is too late. For that reason most zoo medicine is preventative in nature.
Are they any specific diseases or illnesses that you worry about as a zookeeper that you can get from the animals (extra vaccinations, precautions, etc.)?
Again there are quite a few. some of the most common - Salmonella, Leptospirosis, Psitacosis, Hookworms, Lyme Disease, Rabies, Toxoplasmosis (Pregnant women usually aren't allowed to work with cats because of this one). There are vaccinations for some of these, but quite unlike zoo medicine, people medicine is often reactive rather than preventative.
Have you ever seen an animal go crazy (i.e. elephant breaking free from a circus type deal or tiger mauling an employee, etc.)?
never seen it personally, but I've heard second hand accounts.
Of the dangerous animals (lions, tigers, bears, hippos, etc.), which would you choose to have to be placed in their area with for a short time if you had to? Which would you be most afraid of?
Yikes. Can I choose venomous snakes? If so I'll go with that one. At our zoo, I'd probably be most afraid of being in with the orangs. Towan is crazy and scary.
Which animals does a change in weather from their natural environment cause the most problems?