Engineering in general (for even real ships) has to be notified in some fashion of what the bridge is intending to do as far as power to the engines. Otherwise you call for a speed and the ship cannot respond adequately. This produces a natural tension for plotlines, as the chief engineer and captain must be in communication with each other as to what state the ship's engines and power production. A captain issues orders, an engineer knows what the ship can actually do
In TOS (more pronounced in the movies), Scotty would sometimes play fast and loose with his estimates to get things working, as this TNG scene plays out
Navigation in general is directed from the bridge, but in Star Trek you have (more in the later series than TOS) fly by wire, where the computer can accept navigational inputs from qualified crew members from anywhere on the ship (in theory anyways, as I don't know of any situations where the helm was controlled solely by remote command). With warp drive you would probably want computer control doing the driving anyways, since a small navigational error could put you light years off course.
TNG was the series that created the engineering console on the bridge (Voyager and DS9 would feature something similar) and it was believable in a computer driven environment. Presumably the chief engineer does most of his work on a computer console, and thus can work anywhere (it also helps with the drama element).
To your question about why you'd want to steer from the bridge, generally the bridge is the central hub of the ship's operations. You can see everything about the ship's operations from the bridge. That's important in issuing orders so they can be executed by key members of the crew in a timely fashion. For instance, tactical is also on the bridge. You'd want to coordinate the movement of the ship with any weapons you're firing, as well as to avoid any incoming fire. Communication is also there, allowing you to coordinate movement with allies.