As I understand it, the Large Hadron Collider's function is to throw particles into
each other while avoiding hitting the nucleus?
No. In the nuclei are the quarks and trust me, we want to beat the crap out of them (the quarks) with as many direct hits as possible.
If quantum mechanics dictate the position of a particle can only ever be an educated
guess how accurate or reliable is the collider's standard results for a successful
observation? Does it miss more than it hits?
As far as the uncertainty principal goes, its doesn't forbid you from creating states that still have very small uncertainty in space and momentum. In the case of the LHC the wavefunction of the protons is still fairly localized to a small peak that they send around the collider, outside of which there is a negligibly small chance of detecting the proton. So just because the proton is quantum mechanical doesn't mean there is an equal chance of detecting it anywhere in Geneva.
Furthermore, what little uncertainty in the proton wave-packets there is, it isn't much of an issue as far as I can tell. Most of the time something is making contact when the beams cross. The real question is, how often does something interesting happen? It turns out not often - they don't bother recording something like >%99 of the data simply because nothing interesting happened. That is, an automated quick check of the data is performed to see if something triggered an interesting channel, if it doesn't they dump that data, and if something (possibly) interesting happens they store the data and look at in detail later all together.
What happens if the nuclei collide, would their impact result in a nasty reaction that
would destroy the collider at the least?
The nuclei collide all the time and the collider is still here. More importantly there is much higher energy processes happening all the time in the universe for a long time and we are still here.