The major difference is that in vapor phase, the water can leave the container (so you can tell it's in a different form), and which prevents it from immediately rejoining the liquid.
In both solid/liquid phases, the water remains in the rest of the bulk. A few molecules of liquid in a solid or a few molecules of a solid in a liquid won't be detectable, and any created will almost immediately return to the thermodynamically favored state.
why do we have water vapors when our body temperature is also <100°C in the first place?
Water exists in different forms in equilibrium. At body temperature, the equilibrium is for vapor until the vapor pressure exceeds about 7kPa. So it's expected for there to be a certain quantity of vapor present.
why not ice exists at T>0°C? [p=1atm only]
Let's imagine that there is an equilibrium between liquid-vapor and between liquid-solid.
When a bit of vapor forms, the vapor can move away from the bulk liquid. There it can be detected on its own (or mixed with other atmospheric components).
If a few molecules were to form bonds similar to that of ice within the water, the same doesn't happen. They remain within the bulk and wouldn't be detectable. Ice is a bulk phenomenon and doesn't really behave like familiar ice until you've got a lot of molecules bonded together in one place, and that's not favored at room temperature.
If liquid water can evaporate into gas at T<100°C, then why not ice turns into liquid at T<0°C
It does. There is an equilibrium between the liquid and the solid forms. Especially near the freezing point, there will be some molecules that leave the bonding and are not part of the ice crystal. But because the equilibrium points to the solid form, they will be only a tiny part of the mass and will not be seen.
In fact, there will still be some vapor as well. Even below 0°C, the vapor pressure is still positive.
Why water droplets form on mirror but not on wall(just an example)?
The finish of the glass makes a collection of even tiny drops visible. The appearance of the mirror/glass normally is dominated by light from specular reflection. Condensation droplets scatter the light, creating more diffuse reflections. A normal wall will already show diffuse reflection, so adding a little bit of condensation doesn't change that. You have to actually see the drops to notice them, and that only happens when the drops get a bit bigger.
If I look at phase diagram of water, under that conditions we actually have 'liquid', while I understand there should be vapor pressure in equilibrium, details from phase diagram contradicts with the fact. What is the meaning of 'liquid' in the phase diagram then?
In that region of the diagram, it is thermodynamically favorable for the liquid phase to exist in bulk. Outside of that region, even with sufficient amounts of water, any bulk liquid will tend to transition to vapor or solid.
can my ice actually disappear completely at T<0°C, just like liquid water evaporates completely leaving empty bucket?
Absolutely. Leave an ice cube in your freezer alone and it will sublimate away. The vapor pressure is still positive, so some amount of it turns to vapor. That vapor is replaced by dry air from the condenser coils and the ice cube shrinks over time.