After long time of operation, the ice at the freezer compartment of the refrigerator looks like this:
https://www.lighting-gallery.net/gallery/displayimage.php?pos=-248989
I operates the refrigerator with the thermostat at its max position, 5, otherwise the ice appears and melts, requiring me to empty the water draining tray.
Why this is happening?

It looks like a refrigerant has leaked away. Look for oily-wet spots around the circuit, as the compressor oil is circulated (as a mist in the refrigerant) along with the coolant and where it leaks, the oil won't evaporate.

Normally the cooling should be limited by the volume the compressor is able to suck in on its intake, so the pressure there is supposed to be high enough so the evaporation happens at higher temperature and is spread evenly over the whole evaporator, so the frosting should be rather even.
When too low on the charge, the pressure drops too much, causing the temperature to drop way lower at the place where the liquid is evaporating (hence the frosting) and it evaporates very quickly, hence only a small part of the evaporator surface is actually cooling (so the frost is only on a small section).

With most fridges such fault on an older machine does not make much economical sense to fix (the work is rather expensive - extracting remaining refrigerant, fixing the leak, evacuating, new charge, all needing special equipment and a qualified mechanic), mainly when one leaking crack usually means some other will form soon back, as generally the tubing material is just fatigued as whole, so the fix will last only for very short while.

This happens after the refrigerator operated lots of days. Before this is happening, the ice is even.

It did that all the time since new or just recently?

Normally, if it cycles (so mainly when there is no heat ingress into the fridge), the ice just keeps melting away and the temperature gets equalized more during the OFF time, so the effect is less visible (unless the charge is really extremely low). The effect caused by low charge gets more pronounced when the thing is running longer time at once and when there is path for the moist air (aka door left ajar, failed door seal,...).

What you may do is compare the duty ratio of the run cycle towards what it was when new: Lower charge means lower cooling power, so it takes longer on time to keep the temperature. Overall energy consumption may not change that much, as the power draw gets lower too (although not enough to compensate the longer run time, so the average energy consumption still increases).
But when all this, also with the icing pattern, corresponds to how it behaved when new, it could be the thing was just built that way and so it is normal.

Now, after defrosting this refrigerator, and operating it for one day or more, you can see that the ice layer is still uneven, and is thicker at the left part of the coil than the right part.
Also @Medved: If leakage would occur, I would smell this, because this refrigerator, uses isobutane or prophane as its coolant.

No, neither with isobutane, nor propane, nor butane (all three are used), you won't smell anything.

None of these materials is smelly on its own, traditionally an artificial smelly substance uses to be added to them in order to allow the easy detectability.
But this method can not be used in refrigeration systems. First the refrigerants must be rather clean chemicals, otherwise the partial distillation may upset the refrigeration cycle, mainly when their boiling point uses to be bellow or close to that of the main charge. If it is way above (for both the high, as well as low pressure side), there is no problem as it remains just as a liquid mist so won't interfere with the refrigeration cycle itself (like the lubrication oil, fluorescent leak indicators used mainly in car AC's, or additives designed to cool down the peak compressor exhaust temperatures so reduce mainly the oil thermal loading).
Plus the commonly used chemicals use to be chemically aggressive towards the equipment inside, mainly the hot spots (the compressor oil; the compressor exhaust gasses use to be pretty hot there)
And even when added, because the refrigeration cycle is in fact infinitely repetitive distillation, these smelly chemicals would end up dilluted into the compressor oil, so they would be cleaned out of the refrigerant anyway, so it would become non-smelly back.

So as a result, hydrocarbon refrigerants do NOT smell at all.
Their leak could be detected only by a combustible gas detector. Or a special fluorescent liquid in the compressor oil, designed to dye the leak spots (the oil mist is normally circulating along the refrigerant, so it will mark the leak spot so it glows green/orange when illuminated by UV). Or a soapy water - gas leaks create bubbles...
With home appliances the charge is supposed to be low enough and the anticipated leak so slow, it should not accumulate to dangerous levels. And large commercial refrigerators are either subject to mandatory regular inspections and the newer models are switching to CO2 (it is safe, even yields more efficient systems, but operate at way higher pressures, so more expensive to purchase equipment and due to the lower volume, difficult to scale down to small low power machines like home fridges or so).

When the coolant of the refrigerator at the kitchen of the medical storage of Carmel hospital, leaked, it caused strong smell.

1) What was the refrigerant there
2) How fast it leaked? At high concentration you may notice the smell, but then the refrigerant would be out and the fridge completely dead within half an hour.

In your case the leak must be slow, because there is still enough to at least partially cooling. And with home refrigerators there is very small amount of it to start with, barely in the 200g ballpark. If that is slowly leaking over period of weeks or months, you won't notice even when the smell marker would be in it.
The heck even ammonia refrigerant leaking such slow is not noticeable and the ammonia is very strongly smelly on its own...

The refrigerant was leaked fast. And It was one of the combustible gases.

Combustible are practically all of the widely used refrigerants, except the very old freons and the modern CO2. Not that highly flammable as the pure hydrocarbons, but still combustible (so the fridge design must prevent sparks, so e.g. using a PTC instead of electromagnetic relay for motor starting,...).
And a fast leak could be smelly, but mainly it would cause the fridge to immediately stop working, which is clearly not your case (you still get cooling, maybe uneven but still cooling, so some refrigerant must still be in the system, so it can not be any large leak).

For your fridge I would rather guess for a very slow leak, by far not enough to be detectable unless you really find the exact leak spot, but still causing insufficient charge after few months.
Or if it has been doing the same since new, it has been just manufactured that way...