Is it necessary to add email address to paper? Short and simple question. Am I allowed to not include my email at a publication?

On a related note, what should I  do if I don't have an e-mail address, or have an e-mail address at my university which will expire when I leave the university?
You should put your email on your paper.  Not doing so means people who want to contact you can't, and it's obnoxious, non-collegial, and extremely unusual in science.

As for what address to use, I see two main paths: "permanent" and "traceable":

For the "permanent" path: many universities have a permanent alumni email address, which you can set up while still there and which is an excellent choice for long-term usage.  If you don't have access to such, you can get a free address (Gmail is a good "professional" choice, but others can be as well; don't use AOL, Hotmail, or others with a poor reputation).  The advantage of this approach is that your address need never change; the disadvantage is that you may not be using an email associated with your primary affiliation.

For the "traceable" path: it is worth recognizing that even when you have a "permanent" position, such as a tenured faculty post, that it may not be permanent.  People relocate away from long-term jobs for all sorts of reasons, and an email address thought to be forever is not.  Everybody who wants to contact you will understand this.  Thus, rather than use Gmail, you might instead just make sure you always maintain an easily identifiable web presence that includes information about your former affiliations and papers.  Thus, when somebody searches for you, they can easily link you and your paper, even if your email has changed.