Can I somehow make CB Activity "personal" under every member and then have an "open" version for all to see to be used more like a forum?
Like a "small wall" under the member and a "big wall" under the forum for all to see?

You can do whatever you like with your streams as they're all tied to their Asset. If you want more streams you can create them using the Activity fieldtype or a Joomla menu item with a menu type of Community Builder > Plugin followed by selecting CB Activity as the Plugin. Be sure to read the description of the Asset parameter carefully as it explains how to use it in detail with examples.

For example if you create an Activity fieldtype and set the asset to "forums" it will display on every users profile and anyone who posts to it will be visible on every profile, which would basically be a shared stream. If you leave the Asset parameter empty then it will be specific to a users profile and to that field so basically a private stream.

Note no stream is exempt from the Recent Activity page as it forces an Asset of "all" which shows all activity across the site. If you don't want this behavior simply don't use that page as at this time you can not exclude assets from a stream.