Heat is not a property of a system. Heat is a process function. Temperature is a property of a system because is a state function. For instance, the state of a simple gas is given by temperature, pressure, and composition $(T,p,N)$.
Temperature is defined as the inverse ratio of variation of entropy $S$ to changes in internal energy $U$
$$T \equiv \left( \frac{\partial S}{\partial U} \right)^{-1}$$
This is the thermodynamic concept of temperature, which is more general than the kinetic concept that you are considering. Regarding your question, part of the energy given as heat is used to break the bonds and when are broken if you continue supplying energy this will increase the kinetic energy of the molecules.
Moreover, kinetic temperature is not the average of the kinetic energies of all the molecules of the object. This average of kinetic energies is the average kinetic energy. The kinetic temperature is defined as $2/3$ the average internal energy per number density.
At the other hand, heat $Q$ is defined for a given process as the internal energy interchanged which is neither work nor due to flow of matter
$$Q \equiv \Delta U - W - U_{matter}$$
Notice that internal energy is a state function and $\Delta U$ denotes the difference between the initial and final energies, but heat is not a state function and this is why we write $Q$ instead of an incorrect $\Delta Q$.
The concept of process function is most easily understood with the example of a lake. A lake has some amount of water, and this can change by evaporation and raining. You can count the amount of water added to the lake by some raining process, but the lake itself does not have any amount of "raining" or evaporation" only some amount of water. Similarly a thermodynamic system has internal energy but has not heat or work.
Yes the core will warm gradually.
Heat transfer in a solid is conduction.
Ice has a known thermal conductivity and will have a linear temperature profile from all paths from surface to center. There will be concentric rings of constant temperature at all times.
It would be impossible to warm just the surface and not warm up the molecules next to the surface.
Surface will warm to $32^\circ\mathrm{F}$ ($0^\circ\mathrm{C}$) and by that time the temperature profile will be established.
The surface at $32^\circ\mathrm{F}$ ($0^\circ\mathrm{C}$) will just migrate inward. The surface will not go above $32^\circ\mathrm{F}$ ($0^\circ\mathrm{C}$) as ice simply melts. As you migrate inward the temperature slope will remain constant. As the path becomes shorter the core temp will increase.
If you raised the temperature up to $30^\circ\mathrm{F}$ ($-1^\circ\mathrm{C}$) and held it there for a while the core would raise to $30^\circ\mathrm{F}$ ($-1^\circ\mathrm{C}$).
Think about a cutting torch on metal. Once you get the surface to melt it just cuts right through the rest.
Best Answer
Yes, what you say is true: if you add heat during the process of a liquid freezing into a solid, there will be no change in temperature. If two different phases are present in a system, then any addition or removal of heat will act to shift the phase balance one way or the other, rather than changing the temperature of the mixture (assuming the system reaches thermal equilibrium quickly).
So, in the example of water freezing, if you add some heat, some of the ice will melt back into water, but there will be no change in temperature of the system until it is either 100% water or 100% ice.