Most of the flight planning is taken care of by the flight's dispatcher, which is an FAA certificated position. The Captain for the flight prints out the flight plan and dispatch release before each flight, and will discuss any desired changes in fuel, route, altitude etc. with the dispatcher.
Enroute changes in the flight plan are frequent, and sometimes they involve communication with the dispatcher over the radio, but usually as long as fuel is not an issue, the Captain is free to make any changes he needs to.
The desired arrival time is somewhat fluid, since ATC-imposed delays or weather can impact it a lot. If the company really wants to delay your arrival at the next airport (such as when thunderstorms have delayed the previous departures), they might impose a delay on the ground before departure, but that's rare. Normally you just take off when you can, make up time where you can, and if you're early and the gate is not available, park the plane on a taxiway somewhere and wait.