Site analysis tools

I've been wondering for some time now, how we can improve the site to
give the users a better experience and a faster flow from question to
answer. In that search, the question of doing a proper facts-based
analysis keeps popping up.

What I would essentially like is to be able to look at the flow that
happens from when a users has a problem till he/she finds a solution
in our documentation (or on our IRC channel or mailing list). We
should, as documentation writers, have some idea of whether our
efforts are fruitful or not, and whether we can improve pages A, B or
C to make it easier for users to search for an answer and find it, but
without some form of log files or analytics tool, this becomes quite
hard, if not impossible.

I would therefore like to propose that we implement some form of
anonymous analysis snippet on our documentation, so that we can figure
out some facts:

What are users generally searching for when they wind up on our pages?
Which flow of content occurs when a user browses through the docs,
looking for answers to problem A, B or C? Do they go through the guide
as we intended for them to do, or do they pick a different path, and
if so, why?
What are people generally reading about? Which pages are the most
popular, and which are almost never touched (and does this reflect our
own ideas of which pages are the most useful in various scenarios)?

I believe that if we had these facts sorted out, we could more easily
work towards improving our documentation and help people reach the