You walk through the sultry Moscow downtown.

You glance at renovated old mansions, mysterious hotels, de luxe car models, flourishing cafés with tables enveloped in greenery from the South, and stylish pedestrians.

“Luxury car models” would sound a little more natural. I only ever really hear “deluxe” in brand names or ads.

The web of the streets is very dense when you try to find your turn and sparse when you try to reach the next street or boulevard.

“Web of the streets” is not in common usage, but in descriptive writing it could be very creative! Usually “intersection” is used.

You feel drawn by the city through the streets to itsthe thoroughfares and the district’s nuclei -like department stores, theatrers and underground stations.

Ive never heard the word thoroughfare before (US). I would say “You feel drawn by the city through the (streets/alleys) to the (main/central/major) roads” if speaking casually. Also it’s a little confusing what you mean by “nuclei-like.” I think you are trying to say the buildings are within the core of the city. “The district’s (core/central) department stores,…” would be more clear. You could also say “…and the department stores, theaters and underground stations within the district’s (nucleus/heart/core).”

Finally, you extricate yourself from this living maze and reach the destination, entering an office with a comfortable temperature, moderate lighting, and muffling carpets.