I have a doubt about the formula of the kinetic energy of the rototranslatory motion (K = 1/2 MV_cm ^ 2 + 1/2 I_cm omega ^ 2): does it come from Koenig's first theorem or from another proof? and if so, when should I use Koenig's theorem?
For a system of particles, the total kinetic energy is the sum of the kinetic energy of the center of mass (your first term) plus the kinetic energy of the motion about the center of mass. This is true regardless of the relative motions of the particles about the center of mass (system could be a gas, not necessarily a solid). Yes, this is part of Koenig's theorem. (See a good physics mechanics textbook for the derivation; such as Goldstein, Classical Mechanics.)
For a rigid body the kinetic energy of the motion about the center of mass due to rotation is your second term. (Again, see a physics mechanics book.)