Hi,
The issue comes from the CSS code:
font-weight: normal !important;
that you have in the file /templates/rt_hexeris/css/rt_hexeris-custom.css
So the problem is not with your template but with your custom CSS.
Some of the icons of fontawesome need a fontweight solid, while others a fontweight normal.
By forcing the fontweight of all the elements of your website you prevent fontawesome to use the correct fontweight for some of the icons, resulting in the icons not displaying.
You would need to modify your custom CSS to exclude the i tags used by fontawesome:
css-tricks.com/almanac/selectors/n/not/