Did the universe cool from the furthest galaxy in or nearest outward?

Answer 1

Neither. Our position in the Universe is not priviledged in any way; it seems unlikely that it would be either the first or the last place to cool down. It is generally believed that the Universe will look the same (on a large scale), no matter where you are located. Specifically in the case of cooling down, the Universe has likely cooled down more or less the same in different places, as it expanded.