The refluxing of methanol or ethanol over magnesium is indeed the method of choice to dry these solvents. 

As far as the iodine is concerned, this is a classical trick, which is also used to initiate Grignard rections. Is is assumed that magnesium iodide is formed, which is soluble in the alkanols. As a result, a fresh, activated metal surface is left behind. 

This might even work, when the magnesium metal is covered with a very thin oxide layer.

Purification of Ethanol
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