Is there any mention of what happens if a two or more people tried to apparate into the same exact spot at the same time, or if a person tried to apparate into a specific point in space where something already existed (like apparating into the spot where a desk is)?
Warning - science, wild guesses and Star Trek terminology follows:
As a start - even if there is no one on the place that you apparate to it is still not "empty". It is full of air - the invisible fluid that we all breathe for living :)
If the wizard apparates directly onto the place where the air is then the effects would be not less disastrous than apparating into something solid. The air molecules will cause a lot of damage on cellular level like being injected into the DNA on all the wrong places leading to genetic diseases. It will be a new one every time someone apparates.
Fortunately JKR provided (most probably unintentionally) a nice explanation about how is this avoided - every time someone apparates/disapparates a loud bang is heard. This bang is most probably caused by the air filling in the vacuum when someone disapparates and the air being pushed out when someone apparates.
So the most probable mechanism to how it works is that actually you appear out of a very tiny "hole" on the place where you apparate to and (very quickly) expand to your normal size pushing the air out.
The "hole" is most probably with infinite small size and there cannot be two "holes" at the same place.
So if you come out of two different "holes" that are very close to each other (i.e. "the same place") the two persons apparating will just hit each other and bounce off. Depending on the speed of the "expansion" there could be also some injury caused by the hit.
Scientifically speaking there are other issues with apparating. For example how does one compensate for the latitude speed difference. But after all - it is magic :)