Объясните , в каком случае мы ставим артикль a, the перед half, а в каком случае после half?

I found in dictionaries that ‘half’ can be used in the pattern ‘half a/the/this/my something’ (as in ‘they live half a mile up the road’, ‘I have to spend half my time reading his reports’ etc.) or in the position of an adjective (as in ‘they ordered a half bottle of wine’, ‘they sold all the tickets in the first half hour etc.’).
Al Blanco, think about the difference between 'a half bottle of wine' and 'half a bottle of wine'

1) 'half a bottle of wine' is half of a whole bottle of wine, and

2) 'a half bottle of wine' is a bottle that is half as big as a normal one.

So, in 1) it's the quantity 'half' of a bottle and in 2) it's something called 'a half bottle'

The use of the indefinite article should be clear.

'A half an hour' and 'a half a kilo' are wrong. There should be no 'a' before 'half'

The case of 'a half hour' or 'half an hour' is different. They can both be used although I prefer 'half an hour'. 'A half hour' would be considering the period of 30 minutes as a unit of time called 'a half hour'.

'A cool half a million' is correct because if you use a qualifier like 'cool' it applies to 'half a million' as a whole and so the article is required.
Hello, I'm from Australia originally and have now lived in the US for half my life ... or half of my life.

Sometimes the use of "a half" or "half of" or "half an" is a matter of the local vernacular.

For example, I say "half an hour", and Americans often say "a half hour". It drives me nuts, but there's actually no difference, just a matter of what most people say in any given area of the world.

Regarding "half a bottle" and "a half bottle" ... what you don't see in the spoken language is a hyphen which should be used in "a half-bottle". A "half-bottle" is two words joined together to make a compound word, so it needs a hyphen. Hyphens are sorely missing in the written American language, and when I read newspapers, books, and anything else printed here, I'm always stumbling over the flow of a sentence because of missing hyphens. It's an ever-evolving language, and I have to get used to it!

Confusing, but "half a bottle" indicates that half the liquid is there, and half is missing - or - if the bottle is broken, half of the actual bottle it is there and half is missing too!

You have a reasonable grip on your "halves" ... but if in doubt again, continue to ask and eventually you will just memorize the particular uses of this word! After all, that's how all of us English speaking people learnt our own language as little children too. It's a difficult language to master, because it breaks it's own rules so often so asking, as you have done here, will be your key to clearing up other linguistic mysteries in the future too! Good luck!