Addendum re ice melting:

One thing I perhaps wasn't too clear on is that as you start adding heat to ice, it will gradually and very linearly starting increasing in temperature all the way up the melting point of 32 degrees F.

But then something odd happens: the temperature stops climbing and becomes sort of stuck at 32 degrees for a period of time. The reason is that this is the phase change point, where the solid water absorbs heat to become liquid water.

However, when the ice does finally melt, it does not suddenly release energy and steeply rise in temperature. No, the water from the freshly melted ice  is but a fraction of a degree over 32 degrees. Thus, you can have both water and ice at 32 degrees: it just depends on which direction you're coming from.

Of course once the ice has melted, if heat is still being applied, the liquid water will continue aborbing heat and gaining in temperature in a very linear fashion, until you reach the boiling point, at which time the same sort of pause in temperature rise occurs, as the liquid water absorbs enough heat to cause vaporization (i.e., the 'heat of vaporization').