Объясните пожалуйста, 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) its the quantity half of a bottle and in 2) its 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, Im 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 theres 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 dont 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, Im always stumbling over the flow of a sentence because of missing hyphens. Its 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, thats how all of us English speaking people learnt our own language as little children too. Its a difficult language to master, because it breaks its 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!