I want to create automatic connections between a user with a "member" profile and another with an "author" profile. Also the connection should be deleted when the "member" profile is removed from the user group.

Yes, there's a Connection auto action type specifically for establishing (and deleting) connections. Profile type doesn't really matter to that auto action type. It specifically works off their user id.

Automatic login should occur when a specific user group is assigned to the "member" user.

Not sure what you mean by this. When should they be automatically logged in?

That's a bit more complicated. If I understand you correctly you have something like the below.

User A (author) has a field where they select a usergroup
User B (member) is edited and assigned that usergroup

In the above case when User B is assigned that usergroup you need to establish a connection to User A. The only real way to do this is for when User B is updated scan through every user in your database finding users with field values matching one of the usergroups assigned to the user. That's doable with Custom User parsing behavior, but again that's complicated as you've to code that entire check yourself.

Can User B be assigned to multiple authors? Does this really need to be done using usergroups (e.g. are you using the usergroups for actual ACL control)?

Why are you using usergroups for this? Are you actually using those groups to control access on your site? If you aren't you should not be using ACL as it will bloat it and slow your entire site. ACL is accessed basically everywhere and on all pages. It should be kept at a minimum and used as necessary. It should not be used as a means to link users together as that can be done using fields.

One way to do this more clearly is to create a usergroup for all authors. Place them in that usergroup then use a Query Select field provided by CB Query Field to query for all users with that usergroup. This will give a dropdown of every user assigned to that group. Next set it to not display on profile edit or registration. Now it will only be accessible in backend (you can alternatively use CB Privacy to limit profile edit access to moderators only). You'd then be able to easily use that field to establish your connection.

I can provide example configuration of the above if it sounds more appealing than your ACL approach. If however you absolutely need those usergroups then your usage becomes significantly more complicated as you have no information stored on the user themselves to identify who they are assigned to since that's not how usergroups work.