The gun fires straight ahead. The electrons it produces are deflected in the desired direction by electromagnets.
I don't think it's possible to create a mechanical way of moving the gun fast enough to achieve the needs of even a low-res TV screen. If it was a physical mechanism, you can imagine it'd break all the time.
Not one using an electron gun. The article is specifically about CRTs. (Of course I know about mechanical televisions; I just didn't talk about them because they weren't relevant to the topic.)
No the gun does not move. The author appears to have misunderstood this. I don't find it likely that he would have used the words "physically moves to point" if he meant that the electron beam is deflected magnetically. Still, great article.