kaze ga tuyoi desu kara, doa wo simete mo ii desu ka?

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Questions & Answers about kaze ga tuyoi desu kara, doa wo simete mo ii desu ka?

Why is used after instead of ?
In this sentence marks as the subject that’s giving new information (“the wind is strong”). Using would make the topic (“as for the wind…”), which can feel more like setting the scene. Here, simply states the fact that the wind is strong, establishing the cause.
Can we omit です in 強いですから and just say 強いから?

Yes. Since 強い is an い-adjective, you can attach から directly to its dictionary form:
風が強いからドアを閉めてもいいですか?
This still sounds polite because of いいですか. However, keeping 強いですから makes both clauses uniformly polite.

What’s the difference between ですから and だから when connecting clauses?
だから attaches to plain forms and is casual. ですから is the polite counterpart, attaching to statements ending in です (or polite noun/な-adjective forms). In formal or polite speech, you’ll choose ですから.
Could I use ので instead of から to express the cause here?

Absolutely. ので often sounds softer or more formal than から. You’d say:
風が強いのでドアを閉めてもいいですか?
Both から and ので work to express “because.”

What is the dictionary form of 閉めて, and how do you form the te-form of ru-verbs like 閉める?

The dictionary (plain) form is 閉める. To make the te-form for ru-verbs, drop and add :
閉める → 閉めて

What does てもいいですか mean, and how is this permission pattern constructed?

~てもいいですか means “May I ~?” or “Is it okay if I ~?” Construction:
1) Verb in te-form (e.g. 閉めて)
2) (“even if”)
3) いいですか (“is it okay?”)

Why is included in てもいい?
The emphasizes “even if” you do the action. So 閉めてもいい literally is “even if I close it, is that good?” Over time this became the standard way to ask permission.
Why do we add です after いい in いいですか?

いい is an い-adjective meaning “good.” Adding です makes the question polite:
いいいいです
Then いいですか turns it into “is it okay?”

How would you say this sentence casually to a friend?

You can drop most politeness markers:
風が強いからドア閉めていい?
Or even shorter:
風強いし、ドア閉めていい?

What’s the difference between 閉めてもいいですか and 閉めてもかまいませんか?

Both ask permission.

  • 閉めてもいいですか sounds natural and polite.
  • 閉めてもかまいませんか is slightly more formal/indirect (“would you mind if I…?”). The nuance is very close, but かまいません literally means “it won’t bother you.”