No, your are wrong. If you are worry about the score of your page, forget it. Not Google scores your page, but the users that visits your page do it with installed Chrome browser, because he "telephones" data to Google. Google itself doesn nothing.
Never ever!
I could think of another and better solution, but again this solution is not feasible with the cache plugin. So there's little point in talking about it.
I see what you mean, but there is nothing wrong. The reason for it is, that Google Mobile test isn't able to completely render the page like a real mobile device. Check your site with your own mobile device and you will see there is nothing wrong.
I am not shure what you exactly mean with "and goes off the page", but I think you mean the page effect if page is loaded first time with mobile device?
WHM and cPanel have nothing to do with crawler, LScache or ESI. That's why it has to be configured in WebAdmin. If you have a paid license you should open a support ticket.
Create a blank PHP file, copy the code from below in this file, place it on your server where WP is installed and run the file in browser. If done post what is displayed:
<?php
echo $_SERVER['X-LSCACHE'];