Are calls made by third-party apps logged? For the first time I installed app that can make calls automatically (sms/call screener). What worries me is this level of permissions -- however I have to no choice really because stock screener (Android 2.3) works only for calls (not sms), and all apps I found require that permission.

So my question is this -- are such calls (made by app) at least logged? So I could check and see "oh, call to Bahamas for $100"? Or there is a catch, and the call is logged, but the app can clear the log after call as well?
If you mean logged by your phone company then yes any calls that go over their network will be logged and billed to you. If the 3rd party app uses VoIP or something similar they won't log it as a call but then you also won't be billed for it except for the data charges.

If you mean logged on your phone so you can check it in the call history then that depends on the application as it is possible for an app to edit your call history.