Learn English – the difference between “a lot of friends” and “lots of friends”

Yes, there is a very subtle difference between the two forms

Would you mind if I smoke a cigarette outside?

is equivalent to

Would you mind my smoking a cigarette outside?

since both phrases refer to the action of smoking. In the second form, when referring to the action of smoking and you want to use the verbal noun (gerund) smoking, you have to use my.

Would you mind me smoking a cigarette outside?

refers to me personally and not the action of smoking.

Would you mind if..? or Do you mind if..? is just another way of saying May I..?

To the extent that there is a difference I would say that "make a case" suggests something particularly compelling about your argument, while "present a case" is more of an objective description of what you will do.

In your first example there is an implication that the reader should re-examine the assassination. If "makes" were replaced with "presents" I would not perceive any such implication and would instead interpret the statement as an invitation for me to review the argumentation and come to a conclusion, whatever that conclusion may be.

This is a pretty informal sentiment on my part though. "Present a case" feels more formal to me and is not an expression that I come across very often. "Make a case" is extremely common. However, I may be responding to my sense of their relative usage frequency rather than a real usage difference. Regardless, I don't think that you would cause any confusion at all using either phrase in any situation, or that a listener/reader would draw a substantively different meaning from one or the other.