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This sentences look uncoordinated for me. I can't understand the situation.

If you wish to give an advise, an admonition for Smersh, the sentences would look so:
В контрразведке, Смерш, не ищут...

Я дал кусок (or кусочек) мяса своей собаке.

However the sentence "Я дал мяса своей собаке" is correct too, if the word "piece" is not implied.

Correct. However I advise you to use construction "У кого-то (genitive) есть кто-то (nominative)" when you speak about animate objects, because the verb "иметь" sounds ambiguous in such situations.
...

Да, наречия не изменяются.

Правильно.

Но в обычной, повседневной речи лучше располагать наречение "очень" перед тем словом, к которому оно относится:
1) Я очень люблю мою собаку (здесь слово...

You chosen really not easy example. Almost all in this sentence can't be translated "directly", literally: you need to use special constructions of Russian language. For example to translate "a tray...

This translation is absolutly correct. However I think in this situation the common idiom (proverb) will look better:
"Они видят соломину в глазу Путина, не замечая при этом брёвен в своих глазах".

1) When you try to translate construction "something of somebody" always use somebody in genitive, case of something depends on a situation, in our situation it's subject, so we should use...

I think "ввёл в употребление" is better than "изобрёл", it looks closer to english "introduce".

I agree!

BTW, I didn't know, that there is "возвратный залог" also. However look at this:

I saw the adjective of prompt, compared them with other adjectives whose roots are nouns and imitated them.[/QUOTE]I see the article "the" before the word "prompt" in your first sentence ("They gave...

Could you explain why you choose the word "быстро" to translate the word "prompt"? I try to understand your algoritm. And as I can see you understand not well when the ending "ся" should be using in...

Китайцы заняли(ся) аэропорт близкий к моей стране.
However my translation for your example is "Китайцы заняли аэропорт рядом с моей страной".

1) Китайцы, стране
I think these mistakes are typo....

Жизнь Джоди Митика изменилась навсегда в 2007.

1) Жизнь
In modern Russian the hard sign ("ъ") isn't used at end of words. It's used only between prefixes and roots, wich begin with the letters...

I wished move "обычно" at my first impulse, however when I tested variants I didn't find essential difference. Usually any place before a verb is good for "обычно". The best variant for me is "Обычно...

Рынок in dative case is рынку. However in this situation you need accusative case too:
Он обычно приносит корзину на рынок.