Hello scenium,
seems to be the well-known conflict between BS3 and MOOTOOLS. In general, it is not adviced to use a BS3 template framework. As far as I know, the only framework which has fixed the issue out-of-the-box is latest JA T3. (no warranty!)
You might find more infos and a possible workaround in the following topic:

• Elements with tooltips disappear at mouse hover ended

Almost all templates for Joomla developers are already using BS 3. Many general disconnect mootools and only developers are actively using Cobalt mootools and make the product not compatible with BS 3. Why? Why Cobalt can not be used with modern templates or frameworks?

scenium

I feel your pain... but that is the unfortunate history of the Joomla v3 project. It was designed to work with BS v2 incl. some modifications! And it still has MooTools dependencies (even in 3.4.3!)

Workaround & Solution!

A Bootstrap v3 + Mootools fix can be found in Joomlart T3 framework and/or Base template!
You should have a look into the script.js file. It might be located in 2 places, choose one of it:

YourJoomla -> plugins -> system -> t3 -> base -> js
YourJoomla -> plugins -> system -> t3 -> base-bs3 -> js

This file includes several fixes. So, you might load/include the whole file. Or just use a reduced version of it, which includes the related Mootools and Tooltip portions. Easy to identify by the comments. Just make sure to keep the jquery/javasript syntax intact.

You can also copy the code into the HELIX3 template style settings -> Custom Code -> Custom JavaScript
I have made some quick tests with latest HELIX3. It seems to work! ;)

CSS layout issue & images

I have found 1 additional issue with HELIX3 and images/icons: They use a CSS rule of display: block; for all img tags?! This rule has an impact on ALL Cobalt images and icons. The result is a very strange allignment and broken layout. Not sure why and if this display style has been added. It is not a BS3 default!

Anyway, you can fix it and override this rule...
either in your custom.css ...
or in Template Style -> Custom Code -> Custom CSS... or in your Cobalt template CSS...

display: initial; should work fine. It resets the display to "inline". IMHO, "Joomshaper" should change this in an update.

BS v2 to BS v3 ?

You might also go one step further and try to change the bootstrap v2 classes on-the-fly ... ;)
Do you know Joostrap templates? They had an additional update.JS file, which translates the class names.

Also found a demo setup for NoNumber.nl ReReplacer! (maybe not recommended as a permanent solution, but a nice way to test the general behavior and results. ;)

pepperstreet, thanks. I will try T3 framework.