I have a couple of papers that are works in progress or in the submission process. Can I list under my publications list (and clarify they are not yet published but on the road to it)?
There are 4 different stages of a paper:
In preparation. In maths, this means: we know it's true and we're writing it down. In engineering it can be: measurements done. Etc.
Submitted. That's clear what it means. I'd just add: In general, you don't say to which you have submitted it, just in case it got rejected and you re-submit it elsewhere. However, for an application, I would include this information to show the level of journals or venues you aim at.
Accepted or To appear or In print. You've got a final "ACCEPTED NOTIFICATION" from the journal. Maybe the proofreading wasn't done or whatever, but now it's clear that the paper will be published (well, it need not, but that's a borderline thing). I personally prefer to write "accepted" since that's clear. Both "to appear" and "in print" may indicate that the volume and pages have been assigned, which need not be true.
Published. It's published.
You can include papers in all 4 categories in your application. I would actually say that for 2,3,4 you ought to include them. For the ones in preparation, consider the significance of the paper, it can look like: The applicant hasn't done anything yet on it and is just blowing up his publication list. Of course, you have to say in which stage the paper is (well, not for stage 4 papers).
Yes, this is done routinely; provided, however, that these papers are clearly marked as "currently under peer review" or "submitted for review" and specify the journal. If the paper has passed peer review and us merely awaiting publication, it can be marked as "to appear." If the paper has not passed peer review, it should not be labeled as "to appear."
Best regards and wishing a swift publication process.
Probably you'are asking for a PhD application in the US, but, just in case, let me give you an answer from a different country. In Italy, in general, unpublished papers don't count for an application, whatever the position you're applying to: PhD, researcher or professor. So, you can list them, but it's typically useless: they will be discarded by the application board.
More specifically: accepted papers might count, typically not as much as published ones, but you have to attach the acceptance letter received from the journal's editor; submitted papers rarely count; and manuscripts in preparation never count.
In any case, what counts and what doesn't count is specified in the announcement for the applicants.