Thanks for your questions.
I think you could go either way with the separate parts. Using bar-over-bar format is perfectly acceptable, labeling the parts a and b at the margin, as would be separating the parts completely and doing section-by-section for each individually.
When written on the same staff, you'll have to decide if they are separate enough to warrant separate parts instead of using in-accords. And again, utilizing bar-over-bar would be fine, as would separating them completely. I would remain consistent throughout the method book, though. Choose one way and stick to it! (And let the reader know which you are doing ahead of time.)
And yes, I would most likely write out the parts instead of using the parallel motion device, in order to help the student learn to read their music. The fewer complications in the early method books, the better, I'd say.
Hope that helps!