I would never use a sentence like "το ποτήρι του χυμού" to talk about a glass of juice. You're right about "the juice glass"... Commonly, you would hear "ένα ποτήρι χυμό" (a glass of juice), "το ποτήρι με τον χυμό" (the glass of juice), "αυτό το ποτήρι με χυμό" (this glass of juice), "πόσα ποτήρια χυμό;" (how many glasses of juice?) . I'm sorry if it seems too confusing, but that's the way graecophone people commonly speak!
No, that is not correct. It is "the juice glass" that's the name. Don't try to twist it to agree with the Greek. Get used to the idea that languages do not always translate word for word. use the Drop Down hints to find the word required for the translation and you'll be sure to have a truely natural use.
This seems to be another multi-service sentence. "Το ποτήρι του χυμού" is both the glass used for juice but also "the glass of juice". This is making me think of rather comical exchanges in Greek over which it should be. But as with all languages, it works out in the end. Thanks again.