The first load of your page is slow, but subsequent loads are fast (because of caching, I know you know). I noticed that the WP commands we use via WP-CLI aren't working which means you have some code in your site that's conflicting. Generally that's a sign you have a plugin (or theme setting) that doesn't love our PHP, and that would totally slow your site down!
You have a test page I saw, and that loads pretty darn fast, so I think the issue here is that your WP site is not set up as well as it could be.
Change PHP back to 5.3 - Yes I know 5.4 should be faster, it IS for most people, but not all. This may be the issue.
Also try with some of your plugins off. p3-profiler and wordpress-popup are the first two I'd try.
I saw you have WP Super Cache installed, but you're using the PHP cache instead of the .htaccess one. I'd switch over, as the .htaccess is faster. Based on this at the bottom, I'd guess you didn't configure it right for your setup, so you may want to reset the settings and try from scratch (this isn't that horrible, it's a fairly simple plugin).
<!-- Dynamic page generated in 10.031 seconds. -->
<!-- Cached page generated by WP-Super-Cache on 2014-01-16 16:52:09 -->
<!-- Super Cache dynamic page detected but late init not set. See the readme.txt for further details. -->
<!-- Dynamic Super Cache -->
The last though I have is to TRY PageSpeed. It may work, it may not, but it should do some proxy caching and minification if it does.