Can I submit an abstract for an unwritten paper which will form part of my PhD?
I think norms probably vary by field.  In my experience, this would be totally fine.

I work in a business school, and the expectation is that various parts of the dissertation will be published both while and after you work on the PhD.  Doctoral students also present intermediate results from their dissertation work at conferences to get feedback.

However, do be sure that the conference does not take copyright for the work presented, or you may tie your hands in terms of publication later.