Before to dig in, I’d suggest configuring W3 Total Cache not to cache any page accessed from a Mobile Device. If you do cache these pages, you’ll end up serving cached mobile pages to desktop views as well.
That’s exactly what’s happening on your site at the moment. I visited it from my desktop, and got a cached Mobile page.
You can find out how to disable cache on mobile devices here:
https://www.ads-software.com/support/topic/wptouch-w3-total-cache-must-disable-browser-caching?replies=31
Once you’ve done so, you’ll want to flush your cache in your W3 Total Cache options.
We can then look at Google PageSpeed Insights and its results. Before we do so, it’s worth noting that these results are recommendations. You won’t be able to implement every recommendation, and that’s okay. You’ll need to review each file, find out what it is used for, and decide whether it’s worth keeping it (because you’re interested in the feature) or if you should remove it (because you think the performance gain outweighs the feature).
Jetpack’s Mobile Theme inserts 1 Javascript file into the <head>, jQuery. That file is necessary later on, for the Mobile Theme menu to work properly.
Looking at your site, I see other plugins loaded on Mobile: it’s likely that at least one of these plugins (woocommerce-mobile-case-designer, or fancy-box) also needs the jQuery file, and would call it even if you decided not to call that file via Jetpack.
jQuery is used by many plugins nowadays, and it’s probably not something you want to remove from your site.
I’d consequently suggest that you look at the other files mentioned by Google Pagespeed Insights, find out what they’re used for, and decide whether you need the plugin or not.
I hope this helps.