Il cielo si schiarisce lentamente dopo la pioggia.

The si here marks a mediopassive (or intransitive) construction rather than a genuine reflexive. In Italian, adding si to some verbs indicates that the action happens “by itself.”

• il cielo si schiarisce = “the sky clears up”
  The sky isn’t acting on itself; it simply undergoes a change.

Without si, schiarisce becomes a transitive verb, expecting a direct object (e.g., “schiarisce i capelli”).

• Il cielo schiarisce… would sound incomplete or imply the sky lightens something else.
  To express “the sky clears up,” you need the si to turn it into an intransitive/mediopassive form.

schiarire means “to lighten/brighten.” It’s a third-conjugation (-ire) verb with an -isc- pattern in the present:
• io schiarisco
• tu schiarisci
• lui/lei schiarisce
• noi schiarimo
• voi schiarite
• loro schiariscono

By contrast, chiarire (without the “s-”) means “to clarify” or “make clear” in the sense of explaining something.

Adverbs of manner in Italian normally follow the verb:

• si schiarisce lentamente

However, for emphasis you can place it before:

• Lentamente il cielo si schiarisce dopo la pioggia.

The meaning stays the same; you’re just highlighting “slowly.”

Because you’re referring to a specific occurrence of rain, Italian uses the definite article: dopo la pioggia (“after the rain”).

• dopo pioggia (no article) sounds odd unless “pioggia” is very generic or in a poetic/fixed expression.
• dopo una pioggia would mean “after a rain” in a general or hypothetical sense.

Yes. Italian allows flexible word order for style or emphasis:

• Dopo la pioggia, il cielo si schiarisce lentamente.
  You can add a comma after the introductory phrase, though it’s optional.

Italian can use stare + gerund to emphasize an action in progress:

• Il cielo si sta schiarendo lentamente dopo la pioggia.
  This corresponds to English “the sky is clearing up slowly.”

schiarisce = [skjaˈri.ʃe]

• sch before i = /sk/
• sc before e = /ʃ/ (the “sh” sound)
• Stress on the third-to-last syllable: schi-A-ri-sce