Declaring an emergency is an administrative procedure. Dealing with an emergency is operational action. In other words, they are actually two different things.
If a serious situation develops on an aircraft, pilots will go through checklists designed to deal with the emergency. Most emergencies have some sort of checklists for at least part of the problem that causes the emergency. If there is no checklist, pilots use judgment gained from experience to improvise.
Most emergencies are not matters of life or death, and very few emergencies lend themselves to any type of rescue effort (despite what you see in the movies).
As for declaring an emergency, the main purpose of that is to get others out of your way. When a pilot declares an emergency, the sky is his. He no longer has to follow ATC instructions; instead, he tells ATC what he plans to do, and ATC clears a path for him. Other aircraft steer well clear of the aircraft in trouble, either on instructions from ATC, or simply out of a desire to make things easier for the aircraft in trouble.
Pilots and air traffic controllers treat emergencies with the gravest seriousness, and declaring an emergency sets many people and resources in motion to make sure that an airplane in trouble can land safely. Skies are cleared, runways are made available, fire and rescue equipment is mobilized, search and rescue teams are alerted. The pilots of the aircraft in trouble are given complete freedom to work on the problem and choose the best course of action, and everyone else follows. It is quite impressive to observe how much everyone works together to save an airplane in trouble.
A pilot declaring an emergency is always given the benefit of the doubt. Nobody questions whether he has a good reason or not to declare an emergency. If the pilot feels he needs to declare, then he does so. There will be plenty of time to analyze his actions later on, but until he's safely on the ground, he does what he wants, and everyone assumes that he has a good reason to declare an emergency.
As I've already indicated, a lot of the time, there isn't anything that anyone else can do. It's up to the pilot(s) to figure things out. At best, maybe a fighter jet could pull alongside and inspect the outside of the airplane (if the gear doesn't seem to be lowering correctly, for example), but ultimately the pilot is alone. That's why nobody second-guesses him and everyone tries to help.
Of course, sailors work together like this, too—they certainly try to help their fellow seamen in trouble. But you can actually get to a ship and rescue people, whereas that's pretty much impossible with airplanes. So it's all the more important with airplanes to bend over backwards to help a pilot in any way you can. And of course every other pilot is thinking to himself that it could have just as easily been himself in trouble.