Yes, but it's more optional. A Komma is often substituted. http://german.about.com/library/weekly/aa031901b.htm
Great! I was wondering about this too. I answered "I don't hear you, do you hear me?" because I figured it was it wanted. But I agree, I would never say it that way. Like you, I would always say "can" to denote an inablity. "I don't hear you" sounds more like childish teasinh implying intentionally not listen. I have been taught to interpert "nicht" in the present tense with a verb as "do not (insert verb)". But I am wondering if the literal translation is closer to the more archaic sounding "I hear you not".
If you have to complete sentences you need to combine them with a comma and conjunction or put a period in between them. "I do not hear you" is a complete sentence. "Do you hear me?" is a complete sentence.
However, there might be a rule that makes this okay, but in formal English writing it is not.
The way they have it hear is more conversational. I was just being picky.
We're learning German here, not English. Nevertheless, there is nothing wrong with "do" being used with "hear."
"To listen" is different than "to hear". Here is a website that explains the difference. https://www.englishclub.com/vocabulary/cw-hear-listen.htm
It's perfectly correct sentence order in Middle/Early Modern English. The reason that the verb goes to the front in the final clause is because it's a question. Nicht, as an adverb, is generally placed right next to the verb/adverb/adjective it modifies, but with declarative sentences such as "I don't hear you" it goes all the way to the end of the clause/sentence.