Apache conf and MySQL conf will help, but imho i think the problem comes from the aplication itself. linkExaminer is some kinda crawler or something like that? then probably it is doing lot of request to the app and probably the app needs to be optimized, probably adding more cache.
I would try to find problematic urls. Having the site in dev environment, i will enable debugoutput in order to see what happens with some urls. I will look specially to the mysql queries needed for generating the pages and also for the time the scripts needs to finish.
that could give you some hints on what you should need to optimize and so, linkExaminer request can be server without needing lot of server resources.
You could use apache's extended server-status to see what URLs might be executing for too long, and see what's wrong with them.
As said by Carlos, chances are that the site is anti-optimized. This is very common, as some people get pissed of by caching as they haven't read the doc, and will then disable caching altogether as it is much easier :-)
Enable DebugOutput, and see how many SQL Queries are executed by some example pages. Report the numbers to us, and we'll try to help you.
I agree with the above comments and would add that after optimizing the site. You should add a reverse proxy such as Nginx or Varnish in front of Apache, at least to serve static files.
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