I think you're close, Mr. P. [have reason might also imply that the speaker just doesn't want to reveal the reason. This could also be expressed, "well, I have my reasons".]
Yes. If you have a definite reason, you're more likely to use "have a reason". In this sense the meaning of "reason" touches on "excuse".
If I had young children, I would have a reason to leave work early sometimes. [The children would be the reason.] Now that my girlfriend doesn't live in Texas, I don't have a reason to go there anymore. [My girlfriend is the reason.