Ships in the Star Trek universe have an up/down orientation, and when two ships encounter one another they are always the same way up. It applies to other objects too. It would be much more realistic if, when they encountered a giant disembodied floating head in space, it wasn’t always in the same up/down orientation as the ship. In order to be scientifically accurate, giant disembodied floating heads might be discovered in any orientation.
They have a Galactic coordinate system with reference to fixed stars. It would be impossible to navigate through the Galaxy without one.
Might be difficult to communicate with one, unless you’re looking it in the face.
If you’re stopped in space, you don’t want to present your beam (or spine, or belly, or stern) to another (potentially hostile) vessel unnecessarily. It makes perfect sense to position yourself bow-to-bow.
There was at least one episode of ***TNG ***(I don’t remember which) where Riker ordered the helm to maintain a synchronous orbit over a planet’s north pole. He made a lot of mistakes like this, e.g., giving temperatures in “degrees Kelvin.”
Yes, and I suspect that with their level of technology, it’s easy to determine your motion relative to them with speed and precision.
Upsidaisium.
Hell, the Apollo astronauts were doing this back in the 1960s. It proved especially critical during the Apollo 13 mission.
Maybe it just means “drop out of warp and drift” if there’s no point of reference? Like “stop the engines”, not “come to a complete stop”?
At the end of the Doomsday Machine, Kirk orders minimum headway while they make repairs to the warp engines. I can see this on the water where it might maintain stability, but it makes no sense at all in space.
Not even if the engines are damaged? :dubious: 
The definition given for a polar orbit is misleading. If you examine the diagram for the Nimbus satellite carefully, you’ll see that the 22,300 mile altitude was needed to provide a full-hemisphere view of the Earth. In reality, a polar orbit can be at almost **any **altitude above the atmosphere, depending on the spacecraft’s mission.
You would turn the engines off if they are damaged. But then you’d just keep going at the same velocity. The idea that you would make “minimum headway” is the issue that the OP raised - relative to what? Unlike a ship in the ocean, there’s no friction to slow you down when you turn the engines off, and there’s no ocean to be not making headway relative to.
Although, back to the OP, it’s worth noting that the Cosmic Microwave Background does define a particular reference frame, and one that’s detectable by observation. I don’t know if that’s something you can fanwank into it.
To further amplify, the “All” part is because, unlike merchant ships, naval ships commonly have two, four, or even six screws. The ship can be turned faster with an order such as, “Port ahead full, starboard back full.” This is why there is a telegraph for each screw.
Speaking of all stop, in [del] The Enemy Below[/del] Balance of Terror, why does everyone on the Enterprise have to be vewy vewy qwiet while waiting out the Romulans? They could have the largest rave in the Alpha quadrant and the Romulans couldn’t hear it if they were five feet away.
But they really should turn off the flashing lights.
Because, clearly, everyone in Balance of Terror realized that they were actually in a WWII submarine combat story. 
I fan wank that by saying that the other guy’s sensors can pick up the minute vibration of the ship’s surface due to internal noise.
Why run the engines at all? And the impulse engines were fine, not that they were going to get them anywhere interesting in a day.
The remaster shows the rubble of the destroyed planet around the Enterprise, so that could be a reason to get out of there, but he didn’t say that.
That too. If I needed a reference point, I’d use the black hole in the center of the galaxy which maybe could serve as a reference point, its mass being so big. But I don’t think we knew about that in 1966. We didn’t know about it when Niven wrote “At the Core.”
And BTW, the submarine combat story it is based on is a great movie. Check it out.
I didn’t realize that it was based on a particular sub story.