Good job, Sergej! Some feedback for you:

Notes about Moscow public transport. I would really appreciate (1) any corrections.

I would like to write some sentences about Moscow public transport. The underground metro system is highly developed. It has plenty of stations. Each station has a unique design, and that is why many tourists take photos (2) while exploring (3) Moscow's beautiful underground system. During peak hours it is always overcrowded, but using a metro train is one of the most reliable ways to be on time.

Using other public transport does not guarantee you will be on time, because of the traffic jams on the roads. There are dedicated lanes for buses in Moscow. But the bus lanes often are used by car drivers during peak hours despite penalties for doing so. The number of traffic jams decreased/dropped (4) for the last two years mostly because of high parking fees (5) in the City and increased towing of parking rule violators (6).

Notes:

(1) Or: I would really be grateful for...
(2) Definite article not needed - either "take a photo/picture" or "take photos/pictures".
(3) Beauty tends to refer to the state of being beautiful. When referring to a beautiful person/thing/place/etc., it's limited mainly to informal speech: That's/She's/He's a beauty.
(4) became smaller is ok semantically, but would sound unusual to native speakers (US English).
(5) I think it helps to explain that it's not just paid parking, but the high cost of paid parking, that helps deter drivers.
(6) Your original meaning is clear; I rewrote it to be more natural sounding. That is subjective, of course; if I misinterpreted anything, please clarify with additional comments.

Notes about Moscow public transport. I would really appreciate for any corrections.

I would like to write some sentences about Moscow public transport. The underground metro system is highly developed. It has plenty of stations. Each station has a unique design, and that is why many tourists take photos and explore the beauty of the Moscow underground system. During peak hours it is always overcrowded, but using a metro train is one of the most reliable ways to be on time.
Using other public transport does guarantee you to be on time, because of the traffic jams on the roads. There are dedicated lanes for buses in Moscow. However, the bus lanes are often used by car drivers during peak hours, despite the penalties for doing this. The number of traffic jams have been reduced for the last two years, mostly because of paid parking in the City, and an intensive tow truck service for the violation of parking rules.

I would like