Thanks. This was informative. Based on the particulars of my theme I decided not to use your last method because it was "too" particular. But may be helpful in the future.
Some other notes for future viewers of this thread. My theme is a bit wacky and I did a few interesting styling things to fix problems:
IN GENERAL:
I wrapped each .wpcf-form-select in another class, so I could individually style each one.
TO STYLE A SUBMIT BUTTON:
HTML:
<div class="dealsearch-submit">[wpv-filter-submit name="Search"]</div>
CSS:
.dealsearch-submit input { … }
MY THEME WAS GETTING DOTTED LINES ON SELECTORS – SO:
/*Remove Dotted Lines on Selector from Firefox*/
select:-moz-focusring {color: transparent; text-shadow: 0 0 0 #000;}
/* Remove arrows from dropdowns on some browsers */
select {
-webkit-appearance: none;
-moz-appearance: none;
text-indent: 1px;
text-overflow: '';
}
Hope this helps someone!