Rather than put an element inside the math, how about wrapping the man in an element whose background you set:
<span class="math-highlight">
<math display="block">
...
</math>
</span>
and then use
.math-highlight .MathJax, .math-highlight .MathJax_CHTML {
background-color: aqua;
padding: .3em .4em ! important
}
to color it and give it some padding? (Use a <div> rather than <span> around display math.)
If you must do it within the math, then you could use
<math display="block" class="math-highlight">
...
</math>
with
.math-highlight {
background-color: aqua;
padding: .3em .4em ! important
}
to get a similar effect.
No, they don't. For example,
<math display="block" class="math-hilight">
<mi class="math-x">x</mi>
</math>
produces
<div class="MathJax_Display" style="text-align: center;">
<span class="MathJax" id="MathJax-Element-1-Frame" tabindex="0" style="text-align: center;">
<nobr>
<span class="math math-hilight" id="MathJax-Span-1" style="width: 0.758em; display: inline-block;">
<span style="display: inline-block; position: relative; width: 0.584em; height: 0px; font-size: 123%;">
<span style="position: absolute; clip: rect(1.513em, 1000.526em, 2.326em, -999.997em); top: -2.146em; left: 0em;">
<span class="mrow" id="MathJax-Span-2">
<span class="mi math-x" id="MathJax-Span-3" style="font-family: MathJax_Math; font-style: italic;">
x
</span>
</span>
<span style="display: inline-block; width: 0px; height: 2.152em;">
</span>
</span>
</span>
<span style="display: inline-block; overflow: hidden; vertical-align: -0.068em; border-left-width: 0px; border-left-style: solid; width: 0px; height: 0.718em;">
</span>
</span>
</nobr>
</span>
</div>
where the spans corresponding to the MathML elements containing the classes do have the appropriate classes included.
So you CAN use classes to style MathJax output. But you should NOT use styles that change the SIZE of the math, like changes to padding, as MathJax will not know you have made those changes, and will not take that into account when it creates things like square roots, fractions, etc, that depend on the size of their content. (CommonHTML is better than HTML-CSS at this, however.)
On the other hand, you can use MathML to do the entire thing, including the padding and coloring. For example:
<math display="block">
<mpadded height="+.3em" depth="+.3em" width=".8em" lspace=".4em" mathbackground="aqua">
...
</mpadded>
</math>
You can do the same for internal components of the expression:
<mfrac>
<mtext>100 m</mtext>
<mtext>10 s</mtext>
</mfrac>
<mo>=</mo>
<mpadded height="+.3em" depth="+.3em" width="+.8em" lspace=".4em" mathbackground="palegreen">
<mtext>10 m/s</mtext>
This will make sure MathJax knows about the adjusted spacing, so the you can have highlighted math inside roots or in fractions, etc.
Davide