If I am teaching a student a song of Davening such as Shmoneh Esrei, can I teach her the song of the Davening using Hashem’s name? Is the girl allowed to practice the Davening song using Hashem’s name?

If you are teaching a full Brachah of Shemone Esrei, you can teach the song using Hashem’s name if it is necessary. However, it is preferred to say Hashem’s  name as Ado.., without saying the whole name, if the students would understand the intent.

While they “Daven”, they can sing the song using Hashem’s name, but when practicing the song Hashem’s name should not be used.

See here for a similar case.