@klovi,
Quote:1.) There aren't many words beginning with AN "x". -why we used article an not the? we have only one x in alphabet
we'd say "an X" or "the letter X". They are both the same. Each letter X is considered to be a separate item, so it's normal to say "an X".
Quote:2.) I have to clean THE flat before my guests will come. - i thought that with places like, "my" home we don't use article
we say "the flat, the house, the apartment". It's obvious whose home it is.
Quote:3.) I'd rather live in THE country than in A town. - "the country" will be always with define article - it's something like an expression?
Yes, mainly because "the country" is not any particular place (unlike a town). It's a general description.
Quote:4.)
You should try beef at THE Black Cat restaurant.
His pub's called - Paul's Bar. - Why in first sentence we used the and in second didn't. Do we use articles with institutional names?
The answer to this is rather complicated, because it's an irregular usage. There has been lengthy discussion here on A2K in the past on this topic.
For instance, we write "The Grand Hotel", but "Grand Central Station".
Best to learn each one on its own.
And by the way, you need another "the" in your example above:
You should try the beef at the Black Cat Restaurant.
or, You should try the cat at the Black Bull restaurant.