Traveling, Passport, travel documents and returning to USA?

My girlfriend is a citizen of a Middle Eastern country but a legal resident (Green Card Holder) in the U.S.A and wants to travel to Germany to visit her parents. She is not a U.S. citizen yet and her passport from her country of citizenship has expired. Are there any other options for her to travel besides trying to renew her passport from her country of origin? She is not here on Political Assylum or refugee status. I was doing some research on a U.S. travel document but I'm not sure if this would apply to her.

No. She needs to renew her passport.

The US travel document has nothing to do with her expired passport. Germany may let her in on the expired passport with her green card, if so she needs to renew the passport there. If not she can renew it in the US, contact the German Embassy for instructions on how to do that.

If she is a Permanent Alien Resident in the USA
and has her "Green Card"...she should be able to
get her US Passport...if she is in the USA...

If she has been out of the USA more than 6 months
her "Green Card" will be sezied at Port of Entry and
she will be Deported...

She really has no option except to get a passport from her country of citizenship, unless they refuse to issue one to her. If they do refuse to give her a passport, then she may be able to obtain a US Reentry Permit, which is a travel document given to people who are not US citizens but who are refused travel documents by their country of citizenship.

She definitely cannot get a US passport as she is not a US citizen.