Ok...Something I haven't been clear on for a while.

If Mystical Space Typhoon, Dust Tornado or another card such as this destroys Heavy Storm, does it still destroy all other Spells and Traps on the field?

indeed. Its there to stop continuous spells and traps, field cards and face down traps (Though if they are able to be they can be chained to there own destruction). Still dont be discurraged from dust tornado/MST, pretty much every deck has continuous cards and heavy storm to........well........heavy to want to use on just one card.