. . . does ice come to a boil faster than cold water?
No.
Hot, which is arbitrary depending on the application, is acheived by the 'presence of heat.' Cold, which is also arbitrary, is acheived by the 'absence of heat.' Since they are both just arbitrary points on a line, 'hot' and 'cold' are not opposites. Regardless of how cold something is, it can be made colder if whatever heat exists is removed. One is not the opposite of the other, but rather just points on a line indicating the degree of heat presence.
The molecular structure of water at temperatures along the line is not itself a line. It expands when it is boiled, and it expands when it freezes, so it really is an arc. The expansion is really just a molecular reaction, but it has the effect of adding 'surface area' to the molecule. As heat is removed from the expanded molecule which is already consistent with its molecular shape in a frozen state and has increased surface area, the heat present in the water molecules dissipate more quickly. The molecule seems to solidify without the step of contraction and re-expansion.
In reverse, the addition of heat does not affect the 'inside molecules' the same as the extraction of heat does. Though heat can be subtracted through dissipation, it is added through radiation. As the heat radiates, the surface of the ice will melt until it becomes a puddle of cold water, and re-expands as the heat presence factor affects the puddle.
I've never tried it, though, so I could be wrong.
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Afterthoughts: radiant heat radiates from a source; the type of heat transfer in this example would be conductive, not radiant.
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