“Real” dynamite is still available, but it’s not the composition you describe, which was Nobel’s original formula of nitroglycerin, diatomaceous earth, and sodium carbonate. (The standard composition was 40% nitroglycerine, and the level of power and brisance is still used as a relative measure for low-grade high explosives.) The more modern composititon uses nitrocellulose as a stabilizer rther than diatomaceous earth. Because there are more stable and less expensive but functionally compounds, there are only a few facilities that still make dynamite, and there are other formulas which have surpassed nitro-dynamite in popularity. Dynamite, however, is very nonabsorbant of water, unlike PETN or TNT-based blasting gels, and so has application in underwater use like blasting a new canal or water channel. (Composition-4, a very popular military plastique explosive, and similar RDX-based commerical explosives are also very resistant to water, but they are a much higher grade of explosive and not as readily available to the public.)
To the O.P.'s question: virtually all explosives are shock sensitive to a degree–there are a few that require a bicompositional primer or catalyst to make them sensitive, but these aren’t widely used–and the #8 blasting cap has become the standard for detonation sensitivity. A rifle shot on original dynamite formula might very well set it off, and could possibly do so under the right circumstances with newer compositions. I don’t think this would be a reliable way of detonating it, though; unless it is backed up by something hard, as it seems quite possible that the bullet could just cut through the material without making enough of an embedded shockwave to effect detonation. I’d say that it’s almost certain that you couldn’t cause C4 or Semtex to explode in this manner. Pure liquid nitroglycerine, however, can be set off by merely letting a drop of it fall a dozen feet or so (or less if you’ve allowed impurities in the form of the more sensitive nitric esters to form) and could readily be exploded in such a fashion.
In general, commerical explosives are sufficientlly shock-insensitive that even large localized mechanical impulses won’t cause detonation. However, dropping a large mass of it from a great height will likely set it off even if it is allegedly shock-insensitive. As part of a preparation for an air launch test I was peripherally involved in, the Air Force dropped a surplus solid rocket motor to see if the Class 1.3 propellent, normally considered shock insensitive and which will–by unintended experience–take a shot from a high powered rifle without igniting. Dropped from 20,000 feet, it cerrtainly does spontaneously light off with a very brilliant and powerful detonation.
Anyway, the method demonstrated in the movies described by the o.p. just might work, but it would be unreliable, and as usual you shouldn’t take anything you see in cinema for reality, especially the flashy, orange-black mushrooms you usually see emerging from explosions on film.