This only happens at one particular height. At a lower height you will fall down; at a higher altitude, you will fall away from the planet.
I thought at one time that maybe this synchronous orbit was what the Star Trek writers meant by a ‘standard orbit’. But apparently not.
As** Just Asking Questions** implied, The Enemy Below, with Robert Mitchum as Kirk. 
The plot item about the Romulan holding a course for home comes straight from the movie. The ending is different.
They have said “all stop” and “full stop” at times. Both are directions to stop the engines, the latter meaning, do it fast, or we’ll fucking hit a de-cloaking Romulan warbird!
Kurt Jurgens was the U-boat commander.
If the captain orders, “Simon says all stop,” does everyone on the bridge freeze in place?
If that were the objective, they would need to reverse the engines to decelerate to a relative velocity of zero.
When you drop out of warp, you *are *at zero velocity. It’s nothing like accelerating and decelerating in normal space.
As with all things Star Trek, this isn’t always true. Depends who was writing that day. At least one episode for sure, they run up through warp .5… .7… .8 etc.
Just as, for every 10 episodes where the Enterprise is obviously hovering over the landing party, ready to blast something into oblivion with its phasers, there is a Mirror, Mirror where the Enterprise is seen orbiting out of phaser range as a plot point.