1) And what if the monarch is incompetent?
2) And? Does a sense of tradition make him a better ruler?
3) Come on, we're talking about administrating a nation, not tourism.
4) And if that rock is the laughing stock or most hated, it will not change until death, while a President is always recycled based off what the people want, not what a few think is good for them.
5) Hahaha, not even close.
6) Explain how a king is more diplomatically able then a President.

Yes, a constitutional monarchy. This does not mean the monarch has to be toothless, simply that the monarch does not have absolute rule. An ideal balance can be reached.
Then you should have no problem with many European countries current administrations.

I don't understand why you would like to have a government where one person can take away your rights in the blink of an eye.