I've read from dictionaries many many sentences with 'six months', but I've never met with any sentences with 'half a year' or 'a half year'. But on the Internet, one can find counless sentences with the latter two phrases.
Could you please tell me if we can use the three phrases without having to tell them apart? Thank you.
Six months would be the normal way to describe that length of time, but it would not be wrong to describe things in terms of years.
When talking about periods of time of a year or longer you'll see more variation like eighteen months or a year and a half.