Show me mistakes, pleace)

1) I'm feeling that cities are dirty, noisy and expensive places.(I suppose that must be "I feel that cities are dirty, noisy and expensive places.")

Both are correct.  The first one is in present continuous tense to give the sense that you are still feeling that even though this may have been written some time ago.  The 2nd one is in present tense.  This was how you felt when you first wrote down the statement.  However, both meaning is the same.

2) Winters in my city are harsh because it's snowing a lot. (I suppose that must be " Winters in my city are hard because it snows a lot.")  Same reason.

3) City life is becoming more and more expensive. (I suppose that must be "City life becomes more and more expensive.")

4) I'm studying English at a language school in my home town. (I suppose that must be " I'm study English in the a language school at my home town. ") [here the prepositions "in" and "at" are interchangeable. in = inside, at = somewhere around]

5) People generally are believing that country life is healthier than city life.(I suppose that must be "People generally believe that country life is healthier than city life.")

6) The city council is building lots of new apartment blocks in the suburbs. (I suppose that must be " The city council builds a lots of new apartment blocks in the suburbs. ")

7) I'm not liking small towns because there's nothing to do. (I suppose that must be "I don't like small towns because there's nothing to do.") Though the first one is technically correct, the 2nd one is used more often.

Show me mistakes, pleace)

1) I'm feeling that cities are dirty, noisy and expensive places.(I suppose that must be "I feel that cities are dirty, noisy and expensive places.")

You're right, the second option is the correct one: "I feel that..."

2) Winters in my city are hard because it's snowing a lot. (I suppose that must be " Winters in my city are hard because snow a lot.")

I would say it like this: "Winters in my city are hard because it snows a lot."

3) City life becoming more and more expensive. (I suppose that must be "City life becom more and more expensive.")

The correct sentence would be: "City life is becoming more and more expensive."

4) I'm study English at a language school in my home town. (I suppose that must be " I'm study English in the language school at my home town. ")

There is one slight mistake: "I study English at ....". You could use "I'm" but then you would say "I'm studying English at..."

5) People generally are believing that country life is healthier than city life.(I suppose that must be "People generally believe that country life is healthier than city life.")

I'd write it like this: "People nowadays generally believe that country life is healthier than living in a city."

6) The city council is build lots of new apartment blocks in the suburbs. (I suppose that must be " The city council build a lots of new apartment blocks in the suburbs. ")

There are two possibilities: "The city council builds lots of new apartment blocks in the suburbs." or "The city council is building lots of new apartment blocks in the suburbs."

7) I'm not liking small towns because there's nothing to do. (I suppose that must be "I don't like small towns because there's nothing to do.")

The correct sentence is the second one: "I don't like small towns because..."