I like this topic, i am always trying to learn more on the graphic arts. I do think an artist should have one, it might assist when showing clients work.
I look forward to hearing more on this topic.
I'm not sure how to answer the questions (which are all good,) but I know that in my case, I've definitely needed to create a different name. No one ever gets "Lana Gramlich" right (I did a blog post about that,) so I do business as "Eye Candy" now. My logo's among the artwork on my page.