1) Yes, a gear box will shift if there is not the correct amount of oil in it. If there is absolutely no oil in it and a gear has become stuck on a shaft or the shaft stuck, then it won't shift, but that scenario seems unlikely and is refuted by the fact that you can shift when the motor is off. There are lock-out pins in some transmissions that are there to prevent you from doing something like shifting for 3rd gear to reverse. I have not worked on a modern (i.e., post 1986) manual transmission, but it is conceivable that such lock outs could be electronic in newer transmissions.
2) Many cars have a hydraulic clutch that works the "plate" you describe. Be sure that clutch is working. In my experience when the hydraulic clutch went out on my car, once moving, I could still shift carefully. Again, a modern car may have an electronic lockout to prevent you from doing that -- I just don't know. This comment,
i have one that is not selecting gear when the car is on
makes me wonder if your problem is really in the clutch not properly disconnecting the engine from the transmission. If the car is standing still with engine off, put it in first gear, press the clutch in and see if you can start it. Of course that was the way we "jump started" cars many years ago. If the starter won't do anything in the test described, then there is an electrical interlock to sense neutral. How old is the car? Does it have a "computer." If it has a computer, I reluctant to suggest what you can to, as there is a chance it will affect the computer. If it doesn't have a computer, let us know.
3) Even if the clutch is entirely mechanical, there will be linkages. It is still worth doing the test I described in #2 to see if the clutch is properly disengaging.
4) As for your final question, be sure to add the correct amount of the correct grade of oil to the transmission. The fact that you can shift with the engine off tells me the gears and /or shafts are not frozen. They may be damages from lack of oil, but that is another subject.
In summary, my first guess is that your clutch is not properly disengaging or that there are interconnects, either mechanical or electrical, that are preventing the clutch and/or shifting once the motor is running. See suggestions #2 and #3.
What make, model, and year car is it?
John