1. It's a quarter to six.
2. It's quarter to six.
[I think two expressions are correct. Which one is commonly use in the USA?]
3. It is 10:05. It's ten o five. / It's ten oh five.
4. It is 10:5. It's ten five.
[Which one is grammatical, #3 or #4? Do we have to use 'o' or 'oh'?]
It's quarter to six.
Is the way we say it.
10:05 is right
10:5 is very confusing. I might think you mean 10:50
Damon is right -- 2 and 3 are the correct ones.
It doesn't matter whether you write "o" or "oh" -- the sound is the same when you say that time aloud.
In the USA, both expressions "It's a quarter to six" and "It's quarter to six" are commonly used to indicate that it is fifteen minutes before six o'clock.
Regarding your second question, expression #3 "It's ten o five" or "It's ten oh five" is the grammatically correct way to say it is 10:05. The word "oh" is typically used to represent zero when it comes to telling time in the USA.
On the other hand, expression #4 "It is 10:5" or "It's ten five" is not grammatically correct. In order to indicate the specific minute when telling time, you should mention the leading zero before the single digit. Thus, it should be "It is 10:05" or "It's ten oh five".