1. The calendar popup is hidden/cut on tabbed register screen

Your content pane is too small. This should happen with default template too. CSS won't push it above the borders of the content pane for whatever reason (despite a z-index of 999).

2. The pending icon is still "old". The bootstrap icons are limited, maybe icon-time, icon-repeat, icon-refresh, icon-pause are suitable...

Fixed for 1.0.2 already.

3. Regarding BS icons: The online status eye-open/eye-close are hard to differentiate, IMHO. the icon itself is not very clear. Is it possible to color these icons or improve them? (Do you strictly have to use the BS standard set)

Icons seam fine to me, but you can replace them as needed at the below location.

components/com_comprofiler/plugin/templates/bootstrap/images/mini-icons/

I include the entire halfling set as well encase you want to use another bootstrap icon.

4. Field info and help icons
Just made a quick mobile/responsive template test. Not so bad at all! Works pretty well. The only visual issue is on register form... the icons are moving in misc positions... floating around..., if the screen is very small. Should it be fixed for each site template? Or is it possible to improve this in the CB template itself?

It's a simple case of you've too much in too little space. You'll need to adjust the template for your specific needs or increase the content pane. I test default Joomla template and at default parameters only. CB does not yet have the classes for CB to be completely mobile capable, but it will be with CB 2.0 or if needed a later release. Currently I am limited to purely CSS. I can not make HTML changes or anyhing of the source. Injecting jQuery is also not an option. It is common for CB to only load in the templates CSS file where needed.

The issue is in BEEZ20. Also tried the CB DEFAULT template, and the Calendar popup works well.

Please provide a screenshot of what you're seeing with Bootstrap so I know what to look for.

Yes, I am aware of the current state. No offend! Absolutely understand the technical status. I was surprised it worked so well, now Really looking forward to the future.

Yup, really needing more API to properly style some things and be able to adjust HTML, etc.. I've made note for a future CB release (hopefully 2.0) to significantly improve our template usage.