This may be somewhat long but I Hope to be useful..
There are only two major types of energy, namely, kinetic and potential energy. So the answer for your first question**"Does it perhaps turn into kinetic energy as soon as the bonds are broken or maybe alter the structure of the atoms?"** is that the first sentence is correct, and no change in the structure of atoms(the individual atom) occurs nor in the molecules, but only in the crystal lattice structure which is held by inter-molecular hydrogen bonds as you stated.
For the second question, Temperature is a measure of the kinetic energy of the molecule which is expressed in solids by vibration motion, so raising the temperature of, say, an ice cube from -3 to 0 means that the molecules in the crystal lattice vibrate by a larger frequency.
Immediately after reaching 0 Celsius at atmospheric pressure the kinetic energy is not sufficient to make the water molecules to move relatively freely as those of liquid because of the potential energy expressed by hydrogen bonds in the crystal lattice which hold the molecules together (cohesive forces).
To melt the ice cube you need more energy (latent heat of fusion) than that gained by heating to 0 Celsius, but it is not sensible(cannot be measured by thermometer) since it is a form of Potential energy.This latent heat is only to break inter-molecular hydrogen bonds.That is why freezing and fusion point are the same point on temperature scale.Only After complete melting the energy acquired can shift to raise the kinetic energy or the temperature of liquid water.