A unit that is fighting as an overlap is in combat, but if the BG is only fighting as an overlap it is not commited to close combat - i.e. it can move away. Consequently it can have a commander in the front rank as it is in combat.
In the diagram
(4) can't fight as an overlap as it is not side edge contacts with friends who are in front edge contact with the overlapped base - see the definiton of overlap on p75.
If there was a green BG to its right that was fighting then it would count as an overlap.
Must (4) pass a CMT not to charge (1)
Yes - the BG, even if fighting as an overlap, is not commited to close combat so is free to move away - thus it is free to to charge, so if shock troups may charge without orders.
and if they charge: what happens with the joined commander?
The commander remains commited to the front rank - he can only leave if the BG ceases to be in close combat or in contact with routers.
Can (3) fire to (4) with no POA penalty for "BG in close combat"?
If
(4) charges
(1) Then it can't be shot at by
(3)
If it remains where it is, and has a friend to enable it to count as an overlap, it can be shot at without penalty as it would in close combat only as an overlap.
Can (4) use the +2 bonus for the IC that's behind?
Only if it doesn't charge
(1), and doesn't have a friend to enable it to count as an overlap, as in these cases it is in close combat. As pictured then yes as it is not in close combat.