Notifications

You can turn it off within CB Activity > Parameters > General by setting "Notifications" to "Disable". You can have notifications on connection requests, but you'll have to create those auto actions your self as there isn't any system actions for notifications right now. Example as follows would add a notification

notification.42.connection.32

So the above for example would be a notification to user id 42 of a connection request from user id 32. The owner of the activity entry would also be set to 32. These of course in the auto action would be substitutions based off whatever trigger is used (see the system actions for logging connection activity for examples).

So I understand that by turning off the Notifications globe icon and pop-up to check notifications I should still expect connection request emails to be sent, assuming I have enabled that feature?

Also, is there any simple way to enable "likes" to appear in the notifications globe icon/pop-up feature? Otherwise, how does CB notify a user of "likes" made to his/her posts?

So I understand that by turning off the Notifications globe icon and pop-up to check notifications I should still expect connection request emails to be sent, assuming I have enabled that feature?

Yes, the notifications provided by CB Activity have nothing to do with any other notifications. It's just a different form of activity.

Also, is there any simple way to enable "likes" to appear in the notifications globe icon/pop-up feature? Otherwise, how does CB notify a user of "likes" made to his/her posts?

No, as I've already explained there are no System Actions in CB Auto Actions for creating notifications at this time. You've no means of knowing if something was liked unless you look at that something.