CSS z-index quandary

Cd& is right on target.  Before you address anything else, clean up the coding with the validator. Then let's take a look at it.  In cleaning up the code, you may find other errors or that could be the fix.  When you are relying on jquery/javascript it can get messed up or not even work when things are not valid.  

As example, the slider has the code

… disease isn’t <br />germs, genes, or toxins.<h3>It's malnutrition.</h3></span>

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Your h3 tag should not be enclosed by a span.  You can try closing the span prior to the h3.  You may need to update the css to target the h3 inside the slider though if the font no longer matches.

… disease isn’t <br />germs, genes, or toxins.</span><h3>It's malnutrition.</h3>

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When I looked at the theme http://demo.rockettheme.com/live/joomla/cerulean/, it has a couple of minor errors to start, but not the same that your client's page is showing.

Some of these themes tend to use <h> tags for styling.   For SEO, you want to make sure that is not the case.   The coding semantics of your page should read like a highschool term paper.  A main topic sentence (the h1 tag) and several supporting sub topic sentences (h2) tags that each have some paragraphs(p tags) of content.  Inside of an h2 tag, you can have an h3 tag, but you wouldn't want to have an h3 tag at the same level as an h2 tag.

<h1>Dogs Rule</h1>
<h2>Unconditional Love</h2>
<p>blah blah blah blah</p>
<p>blah blah blah blah</p>
<p>blah blah blah blah</p>

<h2>Protection</h2>
<p>blah blah blah blah</p>
<p>blah blah blah blah</p>
<p>blah blah blah blah</p>

<h2>No Liter Box</h2>
<p>blah blah blah blah</p>
<p>blah blah blah blah</p>
<p>blah blah blah blah</p>

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