No and yes - the abundances of elements in the universe is determined by the physics of how nuclear fusion works in stars, and this will be the same anywhere. However, the heavy stuff (basically everything other than hydrogen and helium) tends to collect in the planes of spiral of galaxies where there is a lot of star formation, and not in the galactic nuclei or in the galactic halos.

Yes. Water is not a mineral on Mercury. It is a mineral in certain parts of Earth (where it is cold enough to freeze), but not in other parts (which are above freezing). Water is a very important mineral substance in the outer solar system (in fact it is THEmost important!).

This goes back to your first question: water forms a number of different crystal structures under different conditions of temperature and pressure, and each of these is a different crystal lattice and therefore a different mineral. I know of at least ten of these,and there are probably more. This is part of the reason why there are three thousand minerals described, because you can make as many as ten different types of crystal even with something as simple as water.

How we define minerals is something of an arbitrary distinction. It works on Earth. It won't work when we go to other planets.

Mineral is defined as a naturally occurring crystalline material of definite composition. But this is just a definition. So if there was a planet in which only liquid existed such as molten rock, then there would technically be no minerals... but this is semantic because liquids are also important geo-materials.

The minerals themselves would be the same anywhere. For example,quartz is defined as crystalline SiO₂. If you have crystalline SiO₂,you have quartz. If you don't, either because you don't have enough silicon or oxygen, or because it's too warm to be a solid, then the mineral quartz isn't formed. Even if it's solid, the structure matters: If you melt SiO₂ and cool it quickly, you get silicate glass(like your window) instead of quartz.

So we ask whether the conditions are right for any particular mineral(or crystal) to form on that planet or not. You don't even need to go to another planet to study this, since some parts of the earth(deeper in the ground or closer to a volcanic rift) are too hot to allow some minerals to form. Or if a planet formed around an early star in the universe, it would probably lack heavy elements (anything past iron on the periodic table), so many minerals would not be possible to form. (All the heavier elements were formed by supernovas. We wouldn't have gold on Earth unless it came from an exploding star!)