I have two papers presented at state and national level conferences. These conference proceedings were not published in any journal. Now, should I list these papers in my list of publications or not?
These conference proceedings were not published in any journal.
Were they published at all? If there were no proceedings, then they don't count as publications, because nothing was actually published. (You could still list them as talks rather than publications.) If there were official proceedings in which other people can read and cite the paper (not just an abstract) even after the conference is over, then it's a form of publication, regardless of whether the proceedings appeared in a journal or as a stand-alone volume.
There may be other considerations you should take into account when including this material in your CV. For example, were the papers peer reviewed? Does your field consider them to be archival publications? However, these are more of an issue of how meaningful the publications are, rather than whether they count as publications in the first place.
Depends a little on your field. What do other people in your field do?
But maybe list it as a presentation rather than a publication.
Because in some fields, conference handouts are freely available on the personal websites of the presenters and it's totally acceptable to cite other people's conference handouts.