Heat is electromagnetic radiation - photons, sound is mechanical displacement - phonons.

They mostly propagate the same due to being waves, in most other respects they are very different.

Heat convection is an entirely separate process where heat radiation is aided by the movement of the surrounding medium. Where it would otherwise heat up it's environment, convection keeps the environment from heating up. Compare coffee in a thermos (very little convection) to a cup you're blowing on (significant convection); more air movement - more cooling.

Also, destructive interference does not at all work like that.

Maybe a more useful analogy could be that waves have like walking animations, where in part of the animation they go up, and in another part they go down. Destructive interference happens when a wave in its' "up" phase crosses a wave in it's "down", meaning the resulting movement looks like nothing. The waves don't however interact in any way, and will continue on their way and on their own animation cycles.

The shifting and heating parts are technically true but require very specific circumstances, enough so that I'm more prone to believe it's another misunderstanding of the physics behind this. But I'll give you the benefit of the doubt.