ad 1) I can see the notification icon on the profile, but notifications are never added, even after new comments added by connected users. I have added a screenshot of the parameters in backend.

Ensure CB Auto Actions is installed and published. Next in CB Auto Actions > System Actions ensure you've enabled various notifications. CB Activity itself doesn't log anything.

ad 2) Users should be able to pin an activity; afterwards the activity should stay at the top of the list, but where do I find the option to pin an activity? I expected an option in the menu for managing comments (see second screenshot)

Users are not allowed to pin activity. Only moderators are allowed to pin activity. Once activity pinning is improved to be stream specific in a later release then permission for pinning will be implemented so users in theory would be able to pin posts themselves. As for how moderators pin activity they should be able to use the "Pin this Activity" menu item on a post. Pinned comments is disabled by default for activity comments; you can enable that in CB Activity > Parameters > Activity > Comments > Display.