is it possible to save the registration form midway during the registration process (maybe via an auto action) so that the first field data is saved, but still let the user continue entering data? (so that the 2nd field will see that selection?)

No, for what you're wanting there would need to be ajax present to redraw the edit display of the second field while sending the value of the first field. This is doable as CB has the API to do it, but it's not implemented yet.

Follow-up, if not possible (to save during registration) can it be done on the profile page - i.e. automatically save the users' profile after they change data in one field?

Yes, you could use CB Auto Actions to add jQuery to the profile edit page that posts the form after your first field changes, but this is going to be an extremely odd experience for the user.