It is possible to calibrate both but calibrating the display is much better. Unfortunately, not all displays can be calibrated and so many people resort to calibrating the graphics card. The problem is that this can introduce artifacts causing banding or reduced color-reproduction.

The difference is crucial if you want to see smooth color gradations or even grey scale. Most graphics card have 8-bit per-channel output which gives 256 levels for each primary color (10-bit output has been available a long time but 10-bit displays are only recently becoming common). So when you calibrate the graphics card, the output is adjusted but precision can easily be lost. Imagine, for example, that your screen is too red and that the calibration determines that red must be multiplied by 0.8, then the graphics card could only send values from 0 to 204 (256 * 0.8), so gradations are lost.

When you calibrate the monitor, the mapping between the output and the display is performed on 8-bit output at a much higher precision. High-end displays (even relatively affordable ones) can feature 14-bit 3D LUT exactly to perform this mapping. This is far more precise and allows to render 8-bit or 10-bit color at its full precision.

Keep in mind that when you calibrate a display, it is calibrated to a certain target because the translation from the output to the display is specific. So if you want to calibrate a screen for use with a certain device that does not support loading a profile, you must calibrate it to show colors it expects to display. The entertainment this is often NTSC or sRGB color.

Summarizing to answer your questions:

1. Yes you can calibrate the output but you can also calibrate the display, when supported by the hardware.
2. Yes, generally have to run the software on each device to load the intended calibration.
3. Yes it is possible to calibrate a screen for a device that is not calibration aware but you must calibrate the display to the color-space output by that device.
4. Yes, if you calibrate the output, there can be artifacts.
5. Depends on the screen. If the screen has a builtin calibration LUT and software with the calibrator is able to set it, you can do that. Some screens even support multiple LUT and you can switch between them from the display controls.