Lütfen pencereyi kapat; rüzgâr perdeyi savurup duruyor.

It links two closely related independent clauses and lets the second clause give the reason for the first without an explicit connector. Alternatives:

• Lütfen pencereyi kapat, çünkü rüzgâr perdeyi savurup duruyor.
• Lütfen pencereyi kapat. Rüzgâr perdeyi savurup duruyor.
• A comma alone is common in informal writing, but the semicolon is clearer and more formal for this cause–effect relation.

Because they are definite direct objects. Turkish marks specific/definite direct objects with the accusative suffix -(y)ı/i/u/ü (vowel-harmonized).

• We mean the window and the curtain in this room, so: pencereyi, perdeyi.
• If the object were indefinite, you would typically leave it unmarked (e.g., su içiyorum = I’m drinking water).

It’s the buffer consonant used when a vowel-final word takes a vowel-initial suffix.

• pencere + i → pencereyi
• perde + i → perdeyi This avoids a vowel–vowel clash. The same happens in kapı + ı → kapıyı.

Yes. Lütfen softens the bare imperative and is fine in everyday speech. For extra politeness or formality, use a question form:

• To one person: Lütfen pencereyi kapatır mısın? / …kapatabilir misin?
• To more than one person or formally to one person: Lütfen pencereyi kapatır mısınız?

Imperatives in Turkish don’t need a subject pronoun; sen is understood with kapat. You add the pronoun only for emphasis or contrast:

• Sen pencereyi kapat. (You, not someone else.) For plural/polite imperative, use kapatın (and you can still add lütfen).

Yes: Rüzgâr perdeyi savurmaya devam ediyor is grammatical and means the wind continues to toss the curtain. Nuance:

• …savurup duruyor is more colloquial and highlights repeated/ongoing action, often sounding more emotive.
• …devam ediyor is more neutral/formal and simply states continuity.

Yes:

• Perde savruluyor = The curtain is being blown about/flapping.
• If you want to mention the cause: Rüzgâr yüzünden perde savruluyor or keep the original causative frame Rüzgâr perdeyi savuruyor/savurup duruyor if you want the wind as the explicit agent.

Default neutral order is Subject–Object–Verb:

• Rüzgâr perdeyi savurup duruyor. You can front elements for focus or contrast:
• Perdeyi rüzgâr savurup duruyor. (Focus on the curtain or on who’s doing it.) The verb normally stays at the end in standard Turkish.

Both rüzgâr and rüzgar are accepted. The circumflex (â) traditionally indicates a slightly lengthened or palatalized vowel in careful speech, but many modern texts omit it. Pronunciation tips:

• ü as in German/French ü (rounded front vowel).
• â may lengthen the vowel slightly; without it the word is still pronounced naturally by most speakers.
• The g here is a regular hard g (not the soft ğ).

Both verbs exist: kapatmak and kapamak. In contemporary standard usage, kapat- is the default for closing doors/windows/appliances:

• Pencereyi kapat. is the most natural choice. Kapa! occurs in set phrases (e.g., Ağzını kapa!) and some dialectal/colloquial uses, but for windows kapat is strongly preferred.

duruyor is present continuous. To put the whole auxiliary construction in the past:

• Rüzgâr perdeyi savurup duruyordu. = The wind kept tossing the curtain (was persistently doing so).