Are there Hamilton paths in Cayley graphs of Coxeter groups?

I don't know about Hamilton cycles in this Cayley graph (although someone surely does, and I have a sneaking suspicion that I have heard about them and forgotten). So I'm not answering the question really, but I think this is the answer you want:

To efficiently "traverse" a finite Coxeter group (i.e. visit every element with low memory overhead), then you probably can't do better than the method in John Stembridge's article:

Computational Aspects of Root Systems, Coxeter Groups, and Weyl characters, in "Interactions of Combinatorics and Representation Theory" (pp. 1-38) MSJ Memoirs 11, Math. Soc. Japan, Tokyo, 2001.

You can get it on his website: http://www.math.lsa.umich.edu/~jrs

Look at Section 4. His traversal uses the Cayley graph explicitly, so it will be very compatible with what you're trying to do.

Stembridge has maple packages available for Coxeter group calculations:

http://www.math.lsa.umich.edu/~jrs/maple.html

I don't remember if maple code for the traversal is available on that website.

In fact, for any tree of transpositions in $S_n$ the corresponding Cayley graph is Hamiltonian. Start with my mini-survey with Radoicic which is relatively recent. The type of Hamiltonian cycles you are interested in are best explained in Don Knuth's "Art of Computer Programming", Vol. 4, Fascicle 2b ("Generating all permutations") (preliminary version can be downloaded from the internet archive). See also Frank Ruskey's book "Combinatorial generation".

The answer to both questions is yes. See the paper of Conway, Sloane and Wilks called "Gray codes for reflection groups":

http://link.springer.com/article/10.1007/BF01788686