Despite an idyllic scene, the stroll gives you a few goosebumps.

“Idyllic” is very descriptive! Common in books but not in everyday language. “Peaceful/beautiful/serene/perfect/wonderful” are all more casual.

The scents of herbs mingle in the air, the children caper inat the kindergarten enclosed by a fence of weather-beaten green paint and rust, and a web swing on another playground moves wildly.

“Caper” is also not common everyday language. Normally you hear children “play.” You can also say “frolic” but the word feels a little posh.

“The scents of herbs mingle in the air, the children play at the kindergarten enclosed by a weather-beaten fence painted in green and covered in rust, and a web swing on another playground moves wildly.” This flows better.

It sets a rhythm for the imposing moderato music of the Moscow courtyard summer, immediately muted whenever you enter one of those stuffy buildings.

“Moderato” would almost never be used in casual conversation unless you are studying/working in the field of music. People usually just describe music as slow or fast-paced. If you want to say “moderately paced” you can say that or “evenly paced.”

Imposing also has a connotation that it is “scary” in some way. I think you might mean stunning, striking, grand, or majestic.