We previously used Cloudflare, which obviously fixes most of these problems, but people in these types of circles tend to get pissy about CF.
I'd recommend setting up WP Super Cache (which you're using now) to serve static HTML files (not sure what caching method you are using). Then you need to configure Apache (or switch to nginx) to actually look for those files, otherwise the plugin won't do much (it generates PHP files by default which are taxing on the server). Another option would be to switch to the Cache Enabler plugin (along with Apache or nginx configuration) - static HTML page caching as simple as it gets. Oh and upgrade PHP.
Edit: For Cache Enabler, check suggested advanced configs at https://www.keycdn.com/support/wordpress-cache-enabler-plugi...
Edit2: CF would just by default cache your CSS/JS, wouldn't really solve the issues you're experiencing (most likely heavy PHP/MySQL load due to poorly configured caching). That said I haven't looked into the site that much, might be that you have a lot of portions of the site that can't be properly cached but then again most users shouldn't need to be logged in so doubt it.
I'll look into changing the caching once we're over the worst of this. Forgive me if I don't want to make significant changes during a massive traffic spike. ;)
I mean, most of the WP related assets do not have to be served through PHP necessarily, so they can free up some connections very easily.
If you need help with the webdev side of things to optimize/simplify more on the CSS/JS side, I'd love to help out in this project.