1. The maste comes back.

2. A master comes back.
3. Master comes back.
(Are they all grammatical? Can we use as in #3? In #3, 'master' is used as a proper noun? Am I right?)

4. "Yes, master."
In the expression, 'master' is used in the vocative case. So we don't put 'the' before master. Is it right?
Or can we omit 'an article' before 'master'?

5. When did master lose Chuchu?
6. When did Master lose Chuchu?
7. When did the master lose Chuchu?
(Can we use all the expressions? #7 is no problem. What about #5? Can we delete 'the' before 'master' as in #5?)

1-3 are fine (correct the spelling of "master" in #1), and yes, in 3, "Master" is a proper noun.

4. Your explanation is all correct.

5. ... the master ...
6. OK as it is.
7. OK as it is.

1. "The maste comes back" is not grammatically correct. It seems to be a typo or misspelling of "The master comes back."

2. "A master comes back" is grammatically correct. Here, "master" is used as a common noun, referring to any person who is a master.

3. "Master comes back" is grammatically correct. In this case, "master" is being used as an honorary title, addressing the person with respect. It is not being used as a proper noun, as it is not capitalized. It is common to omit the article "the" before "master" in this context.

4. In the expression "Yes, master," the word "master" is indeed used in the vocative case. In this specific context, it is not necessary to use an article before "master." Omitting the article is common and grammatically acceptable.

5. "When did master lose Chuchu?" is not grammatically correct. The correct form would be "When did the master lose Chuchu?" Here, "the" is used before "master" to refer to a specific person. It indicates that you are referring to a particular individual who is the master.

6. "When did Master lose Chuchu?" is grammatically correct. The capitalization of "Master" suggests that it is being used as a proper noun to address a specific person or character named "Master."

7. "When did the master lose Chuchu?" is also grammatically correct. In this case, "the" is used before "master" to indicate a specific individual, like in sentence 5. It refers to a particular master who lost Chuchu.