Would a non-Pat Yisrael bagel that was toasted by a Jew now be considered pat Yisrael? And what is the status of a pat Yisrael bagel that was toasted by a non-Jew?

Once the bagel is fully baked by a non-Jew, it can no longer become Pat Yisrael. Toasting the bagel is not considered a completion of the baking. However, if the bagel was not completely baked (i.e. slightly raw or very light in color) then the completion of the baking by the Yisrael would render the bagel Pat Yisrael. In the reverse situation, a bagel baked by a Jew is considered Pat Yisrael once the bagel is edible, even if not fully baked. A Pat Yisrael bagel does not lose its status even if a non-Jew finishes the baking or toasts the bagel.