Adverbs – Usage of ‘Quick’ as an Adverb

This is an interesting questiion. What exactly do they modify? The answer is, of course, if they modify a noun, you need to use the adjective, and if a verb, then the adverb. Now, the first sentence, while common, isn't really correct. [Edit: Fumblefingers' comment makes me think twice about that. Anyway...] Look at these:

If we must kill them, let us do so quickly and cleanly, without excuses.
If we must kill them, let the killing be quick and clean, without excuses.

In your first sentence, these two constructions are getting crossed: it actually refers to the killing. However, it is also more concise than us do so. There are times when informal conversation performs this sort of abbreviation at the expense of absolutely correct grammar. In this case, of course, the most concise version is:

If we must kill them, let it be quick and clean, without excuses.

Clearly, the it refers to the killing. Perhaps the reason that the sentence sticks with the adverb is because the noun killing hasn't been mentioned, so there's a bit of an added thought process to mentally fill it in.

This tendency to abbreviate is probably why you will often see this substitution of an adjective for an adverb, especially in AmE:

If we must kill them, let us kill them quick and clean.

You pretty much have to learn on a case-by-case basis when colloquial English will allow this sort of transgression and when it will not.

Yes, that's a nicely written sentence. It's both prettier and more concise to use preferentially here; the equivalent with prioritize would be this:

Music data is given priority over text data so that the system prioritizes the storage of music data when available storage capacity is not enough.

That's ugly, too, since the use of both priority and prioritizes sounds redundant.

I would, however, substitute insufficient for not enough, or do something like not enough to handle all incoming data. Not enough on its own is a bit stylistically inconsistent IMO.