Citing/Attaching unpublished papers in a Research Statement/Profile for a faculty job
Yes!

Your task is to make the strongest possible case for your expertise as a researcher.  Your case would likely be stronger with a few first-author publications, but having compelling first-author work in the queue is certainly better than not having compelling first-author work in the queue.

In fact, even if you have first-author publications, your research statement must describe your ongoing research agenda, in specific, credible, and compelling detail.  Faculty hiring committees are just as interested in your future research plans as they are in your past research accomplishments.  Preprints and submitted papers that support that agenda are definitely appropriate.