If a cop asked you if he could search your vehicle, would you let him?
Not without a warrant.
If a cop asked you personal information would you feel obliged to answer him?
Depends on the circumstance. I got pulled one night and the cop wanted to interrogate me about my whereabouts that evening. None of his business. Give me a ticket or let me go. But there would be situations where I would feel obliged to answer.
Do you believe that a police officer has the "right" to question you about a matter simply because he's a cop?
No. Not just because he's a cop. However, if, while performing his duty as a police officer, he is required to question me about something, then he as the right.