Learn Hardcore Chinese: nǐ kěyǐ kāi mén ma?

可以 (kěyǐ) usually asks about permission or whether something is okay: here it works like “Could you…?”
能 (néng) is about ability/possibility. 你能开门吗？ can also be a request, but it leans more toward “Are you able to…?”
会 (huì) is “know how to” (a learned skill). 你会开门吗？ normally sounds odd or sarcastic unless it’s about a special door/lock or a child learning basic actions.
As a polite request to someone near the door, 可以 is the most natural of the three.

How polite is 你可以开门吗？ How can I make it softer or more polite?

It’s polite enough in casual contexts. To soften or upgrade politeness:

你可以开一下门吗？ (add 一下 to soften)
麻烦你把门打开一下。 (麻烦你 “could I trouble you to…?” very polite)
能帮我开一下门吗？ (ask for help explicitly)
请您把门打开好吗？ (您 + 请 + 好吗 adds respect and warmth)
In service contexts/signs, you might see: 请开门。 (direct/imperative, not “rude” on signs)

Why is there no measure word before 门? When would I use 扇?

In the verb-object phrase 开门 (“open the door”), you’re not counting; you’re just acting on “the door,” so no measure word is needed.
Use a measure word when counting/specifying: 一扇门 (one door), 那扇门 (that door).
In requests like this, just say 开门 or 把门打开—no measure word.

What’s the difference between 开门, 打开, 把门打开, and 开开?

开门: verb + object “open the door.” Also an idiom meaning “open for business.”
打开: “to open (up),” emphasizes the result (now open). Sounds a bit more complete than just 开.
把门打开: uses 把 to front the object; very natural in requests: 你可以把门打开吗？
开开: colloquial reduplication that sounds light/quick: 把门开开 (open it up).
Avoid 把门开 (without complement); it sounds incomplete.

How do I use 把 with this sentence correctly?

Natural: 你可以把门打开吗？ / 你可以把门开开吗？ / 你可以把门打开一下吗？
Unnatural: 你可以把门开吗？ (missing a result/extent like 打开, 开开, or 一下)
Pattern: [你] + 可以/能 + 把 + [object] + [verb + complement] + 吗？

Can I drop 你 or use 您 or address a group?

Dropping 你 is fine when context is clear: 可以开门吗？
Respectful: 您可以开一下门吗？
To a group: 你们可以开门吗？ (asking any of you)

What other question forms can I use besides 吗?

Use A‑not‑A patterns:

你能不能开门？ (ability/possibility; common, a bit more pressing)
你可不可以开门？ (permission/possibility; polite)
你开不开门？ (focus on the action; can sound more direct) All are yes–no questions without 吗.

Can I just raise intonation instead of using 吗?

Generally no. Mandarin doesn’t rely on rising intonation for yes–no questions. Use:

吗: 你可以开门吗？
A‑不‑A: 你能不能开门？
吧 for suggestion: 开一下门吧。 (softer imperative) Rising intonation alone can sound like an echo/repeat question, not a normal request.

How should I pronounce the tones here? Any tone sandhi?

Underlying tones: 你 nǐ (3) 可以 kěyǐ (3–3) 开 kāi (1) 门 mén (2) 吗 ma (neutral).
3rd‑tone sandhi: 可以 is pronounced kéyǐ (2–3). Full natural reading: nǐ kéyǐ kāi mén ma?
吗 is neutral tone; don’t stress it.

Does 开门 ever mean “open for business”?

Yes. Example: 商店九点开门。 (“The store opens at 9.”) In your sentence with 你 + 开门, it clearly means “open the door.” Context disambiguates.

Is 你会开门吗？ wrong?

Not strictly wrong, but it’s odd in normal life (adults obviously can open doors). It fits only if:

You mean a specific skill/technique (e.g., this tricky lock).
You’re talking to a child or being humorous/sarcastic. Otherwise, prefer 你可以/能开门吗？

How do I say “close the door” in the same style?

Basic: 你可以关门吗？ / 你能关一下门吗？
With 把 + result: 你可以把门关上吗？ (“close it and keep it closed”)
关上 emphasizes the result of being shut; 关 alone is fine too.

Where do I put 一下/一开 to soften the request?

After the verb: 你可以开一下门吗？ (most common)
With 把 + result: 你可以把门打开一下吗？ / 你可以把门开一开吗？
一下 and 一开 both soften/lighten the action; 一下 is more general, 一开 feels slightly more colloquial.

What are natural replies to this request?

Yes: 可以。/ 好的。/ 没问题。/ 行。/ 这就来。 (“On it.”)
No/hedged: 不好意思，现在不太方便。/ 抱歉，我现在不能。/ 现在不行，等一下行吗？

Is 开门 a separable verb? What can go between 开 and 门?

It’s a verb–object phrase (a type of “separable” structure). You can insert small elements after the verb:

开一下门 / 开开门 (colloquial, though 开开 usually follows 把: 把门开开)
With 把 you move 门 before the verb: 把门打开/开开/开一下 Avoid forms like 开门一下 (sounds off); put 一下 after 开, not after 门.

Is 给我 okay here, like 你给我开门吗？

你给我开门 can sound bossy/commanding (“You, open the door for me”). It’s fine in very direct contexts or with close friends jokingly.
Politer: 你能帮我开一下门吗？ / 麻烦你开一下门。 Use 帮我/麻烦你 instead of 给我 for requests.

Can I say 你可以去开门吗？ What does 去 add?

Yes. 去 adds the idea of “go (there) and open (it),” implying the door is elsewhere: 你可以去开门吗？ = “Could you go open the door?” It’s natural when the door isn’t right next to the speaker/listener.