with only minor, almost insignificant course corrections
In space, over the long distances, minor course corrections equal major changes at the destination.
the corresponding orbits of the planets would also need to be tracked & entered into some type of mutually available database
Not to mention all the debris associated with the entire path to the destination including asteroids, comets and heavy dust clouds.
But how would it do that the very first time you're given a set of coordinates?
The NavCom system includes long range sensors. It 'reads' parsecs of space ahead of the ship over time. Movement is tracked by the super computer and minor adjustments are made to the course. We do the same thing with our eyes when we drive an unknown road.
The same will be done for space, just not any time soon.
Space does have a coordinate grid system. We also know the general movement direction and speed of major celestial objects.
Every vehicle would have a three dimensional map just like we have today
There would be no need for a map. In Star Trek TNG, the Stellar Cartography room was there for research not for navigation. The computer (NavCom) created the display the people looked at. A NavCom will not need a map, It is a map. A smart map that changes constantly in real-time.
on what axis does that planet orbit and what is its path in order for a nav computer to then figure out where the planet is at any given time.
That is like a glass of water against Lake Michigan. For navigating to a planet in a star stystem that is not our own, you would need to know much more about the system than a planets orbital dynamics. The mass and composition of the star and the destination planet, the movement of the star in its galactic plane (the Sun moves up and down in the Milky Way rotational plane), Moons, asteriod belts, comet paths, CME patterns, revolutionary speed changes of planetary orbits and on and on.
The establishment of a starting time and a time measurement that is universally known. If the nav computer has the starting set of coordinate for a planet and if it knows that it takes 365 days to complete it's orbit then it would also have to know which date the coordinates were established in order to calculate it's current position.
Time is subject to relativity. Computer time not human time would be the defining factor. Your laptop operates on a different time scale than a Cray super computer. We liken it to calculations per second but a computer standard time parameter would need to be set between the computers. What you think of as a second may be hours or days in computer awareness. NavComs would be making minute course corrections to the nano-second or faster. A fractional change now might end up in a destination change of an entire parsec. It may take less than a fraction of a second for a NavCom to determine that a sensed object is rolling away and to the top and that course will intersect you unless a minute change is made now.
There'd be a giant database dedicated to this celestial map. It would contain all map-relevant info, size, mass and movement.
More likely, a set of quantum databases for all explored space. NavCom calls up the dataset for the region of space in its destination. Determines its own exact current location and scans for changes in the data that will intersect the desired course.
If you found a new planet or whatever, your on-board computer would immediately notify all other computers.
Like I said - Quantum
I don't think orbits are circular in reality.
Space is big. The Earth does not have a circular orbit but that doesn't mean nothing does. Space is big. There are many unknowns. Sensor data must be collected in detail for any sucessful navigation. The Voyager Probes are much like rocks that are thrown into space. I find it amazing that they have not collided with anything yet. It just tell me that space is big and very sparsely populated, well, at least here in our parent star's influence. Once they leave the safety of the Sun's influence anything goes.
That was an interesting tidbit on the "time zones" earlier...for a truly universal navigation system that was time-based (to determine positions of bodies), the passage of time would need to be developed as a constant for everyone, regardless of your local time frame reference (how long your "days" and "years" were, for example), would it not?
I think the idea of 'Time Zones' is a misleading term. Time Zones are man-made areas related to position. Time needs to be defined as a duration. Right now, we humans have everything set to the second. Even pico-seconds are based on the duration of a second. The super-computers that will be required to navigate the great distances involved with interstellar or intergalactic travel will need to have their own definition of a standard duration. Perhaps future computer time will be set to the duration of the change from matter to energy or energy to matter. It could be the duration of a single quantum change. Perhaps computers will determine the base time as the duration of a single cycle of the Universe's vibration frequency. For humans, time and speed are relevant. That may not be the case for machine intelligence.
All sensor data is gathered and evaluated as past data. When you see something, it is always how it was not how it is. It takes duration for the information to be registered. This stands true whether it is an image you see or a sensor data being processed by a computer. The stars you see tonight are not where they actually are right now. They are where they were when the light left their surface on its path to your eye.
Time in a different part of the solar system, galaxy, Universe is exactly the same. Mars has a wider orbit than the Earth. We live on the Earth so 1 year is relevant to us. On Mars, 1 year is still a year for us. It is just not a year for Mars. On Proxima 1b, We will still experience a year but a native to Proxima 1b may not even acknowledge their year, day, month, hour or second. They may base their detection of duration on anything.
With the computing requirements that will be required for interstellar navigation it is likely that quantum communication will be common. Thus, no matter where you are it will be the same time as any other place.
If it is 4:30 PM at UTC on Earth, it is 4:30 PM everywhere else. It all depends upon where your quantum base is at. That is how StarDate in Star Trek can work.
Creatures on planets will always think in terms of days, and probably years. Unless everyone on a spaceship grew up on that ship, they would too.
To think that the entire Universe is based on human standards is shallow thinking. We have animals (creatures) on our own planet that do not think in human terms. As a matter of fact, humans are the ONLY creatures on this planet that think in human terms. Does your dog recognize how many days are in a year or how many seconds are in a minute? What about a worm? A horse? a chimpanzee? A dolphin?
One of the things I hate about Scifi Space travel movies is that ships arrive at their destinations oriented exactly on the same plane. The Federation and the Romulans arrive at a planet or a destination in open space and they are oriented the same? This could actually happen but why would it?
Think about it. You just traveled a very long distance to your destination. But before you arrive, you adjust your ship to match the orientation of the other ship?
Now, take that scenario and plug in the navigation concept that your destination is predetermined at the start. What is keeping the Romulans from colliding with the Federation at the destination? The NavCom senses and adjusts for the other ship. Navigation is like a cone. The tip of the cone is the destination and the wide part is the origin. The NavCom sets the destination and uses the funneling effect to get there by constantly adjusting its course.