I tried, "How much is the cost?" and was rejected. I understand this is not a colloquial answer, but it certainly isn't wrong as a translation. I didn't sign up to learn English, after all! I'd be interested in hearing your thoughts on the matter.
I do feel like "How much is the cost" would definitely be understood in English, but I'm not sure if it's it makes too much sense, grammatically speaking.
For example, someone would probably either say "How much is it?" , or "How much does it cost?", or "What is the price?", not "How much is the price?". At least it wouldn't be the first thing to come to mind.
Then again, I'm not native speaker of English. I may be wrong. As for the Greek sentence, it does seem slightly awkward to me, even though understood (just like the case in English.)
Yes, you're absolutely right - "How much is the cost" is definitely not idiomatic in English, I just used it in the lazy literal translation sort of way. All the other examples you give are good idiomatic phrases.