If you use a clean (fresh download) version of the-bootstrap on a fresh WP instance in your test environment and set up your menu structure, can you replicate this there?
Another thing I noted is that Konstantin's example has real links under the top-level menu items, not # links. I don't think this is your problem, but it's another option to investigate if the clean install has the same problem.
Finally, note that on touch devices (tablets and phones, which is where you'd get the responsive menu) menus more than 2 levels deep don't work. For example, when I open Konstantin's example on my phone, I can touch Level 1 and get Level 2 showing, but if I touch Level 2, it collapses and I can't get to Level 3. I can drop Parent Page and get a Child Page to show, but when I touch Child 1, I also end up selecting the Search box because it collapses after I select. Your mileage may vary, but if you're going to have more than 2 levels (e.g. Bowling Fundraiser is 3 levels), what you're getting is actually usable for touch devices, where what you'd get from the correct behavior would not be.
Also, FWIW, your "Welcome to EPX" block has issues on smaller screens. Shrink your browser window to 360px wide and you'll see what I mean.
If you want to use FontAwesome, you may want to consider compiling your own version of bootstrap.min.css from the LESS and replacing the default iconfont with FontAwesome. It's a simple swap documented on the FA site. This would eliminate your FA plugin and let you customize colors, etc. so you wouldn't have to override colors in the child theme's CSS. See bootswatchr.com for a WYSIWYG tool to write the variables.less file.
Hope something there is helpful.