What factors make a sequence evolve non-randomly?
Fay and Wu test is a test that compare expected sequences under a standard coalescent theory model, that is a single panmictic population with non-overlapping generation of constant size $N$ and effective population $Ne=N$, that is the variance in allele frequency at any following generation is $\frac{p(1-p)}{2N}$, where $p$ is the frequency of a given allele. As such Fay and Wu (as well as Tajima's D and other tests) are not only test of selective neutrality but also tests of demographic neutrality as underlined in Nielsen 2001.

It may be hard to make good predictions about the extend and direction at which a given selective or demographic event will affect a given statistic such as Fay and Wu's H. Therefore, in answering the following questions, I am not specific to Fay and Wu's H. Also, answering the questions fully would require making an introduction to coalescent theory which is subject to a whole book rather than a simple post. To fully understand such test, you will need to follow an introduction to coalescent theory. You can have a look at the books recommended in the paragraph General Entry Books to Population Genetics of this answer.


  
  Does negative selection differ from a selective sweep because a sweep is a form of positive selection?
  


Let's first clarify the terms just in case. Reduction in polymorphism at linked loci due to positive selection is called "selective sweep" while when due to negative selection it is called "background selection".

Generally speaking, selective sweep is much stronger than background selection. Also, selective sweep has a higher impact on Tajima's D, then background selection.

To my knowledge, methods trying to disentangle the two processes of selective sweep and background selection mainly use inter-species comparisons to infer what types of sequences are concerned. If The loss of polymorphism is adjacent to a highly conserved sequences and is found in related species, then it is probably background selection at play. If there is a fixed mutation that isn't found in related species and loss of polymorphism isn't found in these related species, then it is probably selective sweep.


  
  After a selective sweep, any rare mutations you see must have come after the selective sweep. Is it because they can't have come during it else they'd 'be diluted' by the positively selected alleles?
  


A selective sweep indeed modifies the Site Frequency Spectrum (SFS, distribution of allele frequencies) to create an excess of high frequency variants (or a relative lack of low frequency variants). See wikipedia > Tajima's D 


  
  Population contraction can create a bottleneck, which makes all individuals come from a common ancestor. But how does population expansion make a sequence evolve non-randomly?
  


Population expansion yield to longer coalescent times and therefore lack of rare alleles. See the introduction of Excoffier et al. 2009 for nice explanation and figure.