If I have an ice cube of, let's say, for example (15$\times$15$\times$15 cm), it was put inside a container of internal dimensions (15$\times$15$\times$15 cm), and the container is strong so as not to allow compression, will the cube melt when put at room temperature? Or will it just melt partially? What will happen exactly?
Sure it can melt. Ice is less dense than liquid water. The difference in volume (the molten water will not fill the cube completely) is filled with water vapor.
In case you are wondering why water vapor is created:
You might have seen the experiment where water starts to boil if you put it in vacuum. If there is no vacuum pump to remove the vapor, the boiling will stop once the vapor has sufficiently increased the pressure. (The equilibrium pressure is called the vapor pressure, and depends on temperature.)
The same thing happens in your example: if the melting ice were to create a vacuum, the water would boil until the vapor had filled it.
All bodies come into thermodynamic equilibrium, and so will the cube with the ice inside, by black body radiation from the environment and convection on the box, so it will melt if the environment temperature is higher than 4C . A relevant answer is here
the height the water has increased due to the melted ice is exactly the same as the height increase due to buoyancy before the ice had melted.
If the outside temperature is below 0C the ice will not melt but cool further, the equilibrium will be a lower temperature
When water transitions from a liquid to a solid (freezes), its density decreases by about 9%, which is unusual among chemical compounds. This is the reason why ice floats in liquid water. When water is in the ice state, it's density starts to increase slightly as the temperature falls further below zero, though it never becomes as dense as liquid water.
So there will be no problem . If you filled the box with water and froze it, there would be great pressure on the containing box due to the 9% decrease in density as the crystals of ice form. The reason for burst pipes in a freeze.