Why do we use Babylonian names for the Jewish months? In addition, if they are not Hebrew words, how can we interpret them as if they are found in the Torah (i.e. the name Elul and all its allusions and others)?

The Ramban in his commentary on the Torah says that the Babylonian names are to remind us of the Babylonian exile and our redemtion from that exile and the building of the second temple. Rabbi Meir Simcha of Dvinsk (in Meshech Chachma) explains that the Babylonian names are preserved to remind us that the Jews in Babylon were culturally assimillated but still devoted to Torah values.