After dragging the users to the group, there is save or cancel option in the bottom right corner of the lightbox. If you can't see this, you will have to zoom out in your browser.
Yes, when you are logged in as a particular user, each change made by that user is tracked. This information can be seen by viewing the log in the change history of the thing, thing template, thing shape, etc, in question.
For further questions on users and user groups, please refer to the security section of our wiki documentation on the community site.