Thanks for your response, Oliver. Twistor space interpretation makes sense (although I am not familiar with it). I will take a look at the references. Many thanks.

This is an extremely clear exposition, Dan! Now that orbifolds are not that scary and I even feel familiar with them. Having understood what orbifold fundamental groups should be, I now try to digest the definition and reproduce what you describe above by definition. Many thanks!

J. Martel, you may be right, but basically I appreciate any comments and any viewpoints. Thank you for the answer, David.

I organized the questions and corrected typos.

Sorry for my late reply and thanks for the detailed answer. I now have better understanding on the problem.