I want the connections tab to appear only to owner's profile and to administrator, how can I do that?
You can use CB Privacy to let the user control who can see it. You can also use CBSubs Fields Tabs Protection and protect it with a plan that no one will ever be able to obtain then set the protection to be ignored for owner and moderators.
Also, is there a way, in the connections tab, to show a determined field besides username and the email icon?
Unfortunately, no, not without editing the Connections plugin.