update
Just to make it clear, I'm answering this part of the question: Are there any known algorithms or problems which actually get easier or faster to solve with larger input?
As noted in accepted answer here, there are no algorithms working faster with bigger input.
Are there any O(1/n) algorithms?
Even an algorithm like sleep(1/n) has to spend time reading its input, so its running time has a lower bound.
In particular, author referes relatively simple substring search algorithm:
http://en.wikipedia.org/wiki/Horspool
PS But using term 'negative complexity' for such algorithms doesn't seem to be reasonable to me.
