Yes, any animal can get sick from viruses, bacteria, parasites, cancers, or even their own malfunctions. Socially cooperating animals like us show their illness and others help out sick individuals. Animals that do not help each other usually don't show nearly as many outward signs of sickness because the other animals may attack weak individuals.

Yes - everything can get sick, even bacteria that make other things sick (those bacteria can still get viruses).

They get illnesses. I don't know if they get "sick" in the sense of vomiting. There are special, aquatic veterinarians to care for fish and sharks.