I know that some adjectives (such as easy & short) can be used as adverbs in some situations, but when can this happen and what adjectives does this apply to?
This definitely works: "He stopped short" But does this?: "He fell pretty hard"
I don't think there's much more to it: some adjectives can simply be used as adverbs too. Some can only be so used in certain idiomatic expressions (your estimate fell short), others in a broader context (she drove by fast). There are also other adverbs that simply don't end in -ly, like soon and yonder.
The now productive suffix -ly for adverbs is relatively recent; Dutch and German don't have it—that is, the suffix exists, but it is used differently. In Proto-Germanic/Gothic, the suffix -lîko- could be used to form adjectives from nouns and other adjectives. (Cf. manly, soldierly, womanly, masterly.) The normal suffix for adverbs was -e in Old English, which still exists in German. When the -e ceased to be pronounced in English, at some point -ly became the normal adverbial suffix (it is supposed that there had been adverbs in -ly that had the suffix because they were based on earlier adjectives with -ly, Oxford English Dictionary on -ly_2). This history of the suffix is probably the reason why we still do not use it consistently today.
A small list of seemingly normal English adjectives that can also be adverbs, to which I invite anyone to add more examples:
Edited: The question remains why these adverbs cannot be used before the finite verb:
They quickly followed her.
They soon found her.
*They fast drove to the palace.
I have a theory: because words like fast can be used as regular adjectives, and because many verbs can be used as nouns, it would be very confusing if we could say both *they hard work and their hard work. It could easily lead readers on a false scent, especially in more complex sentences. That could be a reason why we do not use these words in that particular position.