Not the OP, but having worked on the EMS side of a fire department, I can give some insight (though my department was much smaller than OP's):
1) Usually each fire engine is staffed with 3-5 firefighters. A minimum of four is really idea, but due to staffing/budget issues, many departments only have 3 people on each engine. The crew on each engine is usually one chauffeur/engineer/driver (who drives the fire engine and operates the water pumps), one officer (in-charge of the crew), and one or more general firefighters.
In most departments, each crew of 3-5 works for 24 hours and then has 48 hours off (or 24/72, or something similar), there are multiple groups of firefighters that all work on the same physical fire engine. Today, the "A shift" crew of Engine 1 might be staffing the fire engine numbered 1, but tomorrow, a different "B shift" crew will staff the same engine and operating under the name "Engine 1"
On a somewhat related note, the organizational/command structure of the fire department depends largely on size. My department/county had several on-duty battalion chiefs who each oversaw a handful of engines/trucks. FDNY, on the other hand, has ~50 battalion chiefs, each overseeing ~6-10 fire crews, and has higher organization into ~10 divisions and 5 borough commands.
2) It depends on the type of fire/call. If it's an automatic fire alarm or medical call that comes in 20 minutes before shift change, the out-going crew is expected to run the call to completion. However, if there's a large fire/major incident, the command staff usually coordinates shuttling of crews on and off scene to maintain adequate staff on scene, but get tired crews home and fresh crews there.
3) For any large fire, multiple fire engines/trucks from our department will be dispatched. There are standard protocols that determine how many and what types of units are dispatched for each incident. If the needs of the call overwhelm our department, or if the incident is close to our jurisdictional line, the command staff on the scene can call for mutual aid for neighboring departments. Most fire departments have automatic aid agreements with neighboring cities/counties so that the dispatch centers can talk to each other send the needed units