I know from my own experience, undergraduate research is very important to many schools. As a post-doc I mentored a couple of REU projects and it definitely helped me land my current job.

To answer your first question, "Yes". If you are interviewing at a mid-sized or smaller school, there's a very good chance this topic will come up. It's worth putting some time into thinking about projects.

As for your second question, when applying to schools with more emphasis on teaching, try to work your REU ideas (or at least your interest in REUs) into your teaching and research statements. I'm less sure about applications to R1 universities - Maybe just in your teaching statement?

As for the last question, definitely "No". It's great to have a project related directly to your own research, but not necessary. For me, my subfield is affine Lie algebras and vertex operator algebras (not very undergrad friendly), last spring I mentored a group looking at partial fraction decompositions in Euclidean domains (still algebra, but definitely not my own research). Also, keep in mind they are undergraduates not PhD students. The research does not have to be really groundbreaking. Working out details in a basic example (in say "stable homotopy theory") might make a good project.