Sorry - but this is a Joomla thing, not a Fabrik thing. Fabrik is designed to work with Joomla icons, which is why it works with protostar - if you want to use a different template you need to make sure that whoever provides the template ensures it is compatible with protostar. Unfortunately there are a LOT of templates which aren't.
From a technical perspective, joomla/bootstrap/fontawesome icons are all rendered using fonts. The <i> tag calls the font, the icon class does a css:before to add the correct character for that icon in that particular font.
So, protostar uses a special font with a subset of icomoon. Your template uses the bootstrap glyphicon font. And other templates use fontawesome.
I guess it might be possible to use CSS to select the joomla font for joomla icon classes and the bootstrap / fontawesome font for other icon classes but this risks having icons which don't look like they go together.
So, your choices seem to me to be:
1. Try to fix this yourself - however since you don;t seem to be particularly expert in browser dev tools (which you would need to be to play around with CSS at this technical level), I would not recommend this.
2. Try to get joomlashine to make their templates compatible - which is not going to happen quickly.
3. Try to find an alternative template which is joomla icon compatible.
4. Try to find a template which uses fontawesome but is not compatible, and use my fix to make it compatible.
That's all I can offer as advice. Please let us know how you get on.
S