When someone disapparates from Location A with the intent of reaching Location B, is there any way of tracing their route? (I am not sure if I recall something like this being done at some point or not.) Is there any way of telling where a person is going after they disapparate?
Yes and no.
"But how did they find you so quickly? It's impossible to track anyone who Apparates, unless you grab hold of them as they disappear."
Remus Lupin - Deathly Hallows - chapter 11 - The Bribe - page 205 US Hardcover
So mostly no, but there is that small contingency.
It seems unlikely. There are quite a few cases where it would have been advantageous to do what you describe if it was possible but we don't see this happening. DH spoiler:
When the trio apparates away from the Lovegood's house the death eaters most certainly would have used the power you described to chase after Harry. Voldemort would have gone after Harry at Bathilda's house after they apparated away if he could have.
In the Goblet of Fire:
The ministry of magic officials probably would have gone after the Death Eaters responsible for endangering the muggles if they had the ability to do so.
Now none of this is conclusive that it isn't possible but it is suggestive that it's at least VERY difficult if it is possible. If Voldemort can't do it and the ministry can't do it the only person I can think of that might have the power would be Dumbledore. I don't recall a situation where it would be to Dumbledore's advantage to be able to do this. The evidence points toward this not being possible.