"Her name is Gretchen. That's a German name, right?"
No.
"Gretchen", while totally archaic, is based on a German name, yes.
But this "Krettshn" thing Americans mutated it into is definitely not.
This is the original German pronunciation:
https://upload.wikimedia.org/wikipedia/commons/transcoded/2/2a/De-Gretchen.oga/De-Gretchen.oga.mp3
Also, keep in mind that "-chen" is a diminutive suffix, "Gretchen" literally means "little Greta". It feels very strange for us when the person is an adult.