Should a professor give students a take-home exam when the answers might be available online?
To answer in order:


I don't see why it's unethical for the professor to give a take-home exam. I don't dispute your (implied) assertion that this may benefit those who are willing to ignore the professor's explicit directions -- but that's unethical of them, not the professor, in my opinion.
This is entirely up to the professor. It would likely be within their right to do so, but my guess is that most professors would shrug their shoulders and make a note to themselves not to ask that question on a take-home exam again. You'll argue that this is unfair to students who follow the professor's instructions -- and you're right. Unfortunately, however, the world is not fair. That doesn't mean I -- or the professor -- condone cheating, but the professor likely knows that this risk exists with a take-home exam, and has judged it an acceptable risk.
What is the dean going to do in this situation? Force everyone to retake the exam? This may not be feasible -- for example, at my university, some classes have online sections off students who aren't physically present on the main campus. This precludes in-person exams oftentimes.
My advice would be to use a throw-away email account if you're so inclined. Keep it anonymous, but alert the professor. Understand though that they may choose not to act on the information -- and there likely isn't much you can do.


I don't mean to this be harsh -- just realistic. A better solution for the professor might be to explicitly allow online resources, but then structure the exam such that those resources don't help students any more than memorizing the material would. The easiest ways I can think of doing this are (1) opinion-based questions and (2) time limits that make accessing such resources a costly waste of the available time.