Im calling acoustic levitation crazy beacause it would require quite a calculative mind and vast knowledge in physics. Otherwise, it might work, though im not sure how the movement of air would actualy make it denser. I was implying the use of the wave's sheer power to move onself, as if with air magic. Though of course, that is pretty harmfull to a human being.
As for the temperature control, the outcome will vary depending on the way it works. Wether it is an isolating power that lets nothing through, including physical objects, not only heat radiation, or drains/gives heat to an object, keeping the temperature constant despite any interactions - you decide. Unfortunately, that is all i can think of myself.
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Im not sure, but since burning is actualy the process of oxidation, it may occure without any temperature changes involved if the power forbids that. Though most of the time, heat acts as a cataclyst, so if it happens, it should progress slowly.