When the DeLorean in BTTF travels through time, it instantaneously jumps from one time to another in the same 3 dimensional position. Does the DeLorean contain a spacial history device to determine if there is something occupying the space it would notify the user? If not, does that mean that all uses of the DeLorean when traveling through time have been extremely lucky that they didn't time travel into another object?
Yes, they are really lucky.
I don't believe it does have a Spacial History Device™ based on this bit of relevant dialogue.
Marty: You know Doc, it's gonna be a hell of long walk back to Hill Valley from here.
Doc: It's still the safest plan. After all, we can't risk sending you back to a populated area, or to a spot that's geographically unknown. You don't want to crash into some trees that once existed in the past. This is all completely open country! So you'll have plenty of run-out space when you arrive. (Bolded for emphasis by me).
I don't think the Doc would have to worry about such things if there was already a system in place to prevent such things occurring. Also within the Back to the Future universe time travel is shown to be instantaneous, so it doesn't seem like there could ever be this kind of preventative system in usage(no matter how great it might be).
I think something like this would probably happen.
This answer is totally speculative (and probably not what was specifically intended by the movie's creators), but there's some kind of bright light that appears in front of the DeLorean right before it jumps, as seen at http://fxrant.blogspot.com/2010/08/back-to-future-einstein-jump.html ...so maybe instead of just instantaneously disappearing from one time and reappearing in another, you could imagine it actually creates a sort of hole in spacetime in front of it (like a wormhole, see Stick's answer to another question about the DeLorean here), and drives through front-first. Then there wouldn't be any danger of two bits of matter suddenly occupying the same space and overlapping, though if the other end of the hole opened in the middle of solid matter (the inside of a mountain, say) the DeLorean could "crash" when its front first tried to pass through, as if it had hit a door with a brick wall behind it.
I always assumed some sort of temporal inertia. Much as with H.G. Wells's eponymous Time Machine, the depiction is of them essentially traveling in a straight line through time to their destination. Thus, the car remains roughly attached to the position of Earth/the galaxy/etc and there's the opportunity for it to move out of the way of obstacles before re-materializing.
From a real-world perspective, space-time is one continuous field, so anything that travels (or jumps) through time would have to travel (or jump) through space across the exact same reference plane. Think of it as walking down a road, except when you walk down a road through space you are really walking through time as well: everybody is a time traveller. (The speed at which you walk relative to a fixed point in the road determines the speed at which you move through time relative to a fixed point in time in which the road exists.) Future time-travel already exists: using conventional means one can travel at near-light speed into the future, for instance.
Even if you fold or warp space, all you're doing is bringing two points in space-time closer together: the intervening "tunnel" in the wormhole cannot form if it's interrupted by a medium in space-time through which the wormhole cannot form. (From a quantum tunneling perspective, if the wormhole would deposit you in a solid object, it wouldn't have formed in the first place. To simplify things greatly, particles can "predict" the future to the extent that they choose one path over another prior to getting to the finish-line, which in this case is an observer in the past.)
It therefore stands to reason that the DeLorean travels like a normal car over an extremely (relativistically) foreshortened path in space and time to get to its destination, as indicated by the burning-rubber tracks that shows that the car has velocity relative to the planetary body through the wormhole (or whatever it is).
This also explains why it stays in relatively the same spot on Earth: it never leaves the planet's gravity well, so its inertia remains in the same direction, relatively speaking.
Note: Doc Brown's quote in the accepted answer does not contradict this: Crashing into an object near your destination remains a concern, just like landing on a runway is a concern if momentum propels you past the end, into an object at the end of the runway. We see this in the first film when he knocks over one of the Twin Pines. This happens because the pines grew up in the path of the vehicle in its past, past the point in which he went into the past and began to accelerate again back... towards the future.
There's probably a "feel the way through spacetime" effect. Adjustment to contemporary ground-level across the intervening years, if necessary, which perhaps also explains why departing from a point upon a spinning Earth orbitting the Sun, orbitting the Milky Way, etc, does not lead to being dumped somewhere in deep space, or worse...
It maybe would not/could not directly deal with someone who parks a car across the 're-entry' path of the delorean at or around the hour of the day of destination, and trees are awkwardly 'transient' in position, if not time, but gradual changes (due to alluvial deposition, erosion or some seismic dip-slip earth movement/subduction-riding) are caterable-for and you hope for the best in all the exception cases not thus covered.
It must be fairly accurate, if it works for time-travelling trains (prior to engaging hovermode) seemingly transitioning between historic and future rails where possible, without obvious problems with rebedding.