Does adding additional vehicles to city services increase coverage area?

The service area is always the entire city. Travel time determined whether you'll reach the destination before a building burns down, patient dies, or criminal gets away.

Early on, with little traffic, one single service placement can usually cover the entire city. Adding more vehicles lets you dispatch to more places at once. Once traffic increases, the effective service area is reduced, no matter how many vehicles you add (except for helicopters). You can wind up in a situation where a vehicle can't reach the building next door because it can't even exit the vehicle bay. Good placement, road layout, and public transportation will keep your services most effective.

You'll notice that the game lets you plop the vehicle bays apart from the building. Don't lay them all next to each other so they exit on the same road in the same direction. Place them on opposite sides of the road and across intersections. That way, traffic in one direction will only clog up some of your vehicles instead of all of them.

The land value only benefits in the immediate vicinity of the service building, as the closer the event to the station or a hospital, the faster shall be the response, and the higher the priority is. That is, the closest event is more likely to be served first, save for some traffic boondoggle.

Additional vehicles and other upgrades increase the happiness (or prevent unhappiness) of the people and businesses. It also saves you from loss of income. For instance, if the firemen did not get to the fire in time, worst case scenario is the building completely burns down and has to be demolished (and charred people make fellow citizens unhappy), or it can be saved, but damaged and abandoned, which takes awhile to get repopulated. In both cases the happiness has to re-accumulate from the zero, thus postponing increase in density if that is desired. Above also affects land value in a negative way as much as any abandoned building would.

So many different coverage areas...

Land value (used to attract wealthier sims): The fire station gives a small area land value increase. Attachments and vehicles do not increase this area.

Happiness (used to increase density): The fire station, attachments and vehicles do not directly affect happiness, but a burnt building loses its density progress and starts over from zero.

Fire threat: The building as well as the fire marshal vehicles (not the regular fire trucks) reduce the chance of nearby buildings catching fire. More fire marshals means more wandering fire prevention. See this in the fire services dataview as the green street area around the building and the fire marshals.

Does adding additional vehicles actually increase the service area of these buildings?

No, the regular fire trucks only put out fires. If you don't have fires, you don't need more trucks.

Using your example of a fire station, regardless of where you place it in your city it will always give you full coverage of the entire region. The reason you would place multiple fire stations is due to demand and distance. Demand can be resolved by simply adding more fire trucks, while distance relies on the physical placement of the stations itself.

This applies to all similar plopables, so for example police stations and hospitals.

Plopables that use the 'stop' mechanic, for example schools and mass transit, require stops to be placed which each have a finite range and only offer a service within that range, requiring the placement of stops around your city to provide coverage. Again with these you can expand to meet demand by adding more school buses or bus shelters, etc.