Question

1. Will you close the window?
2. Would you close the window?
3. Can you close the window?
4. Could you close the window?
5. Do you mind closing the window?
6. Can I ask you to close the window?
7. May I ask you to close the window?
8. Could I ask you to close the window?
9. Might I ask you to close the window?
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Are the sentences all possible?
#9 seems to be the most polite sentence.
What about #5? Is it the second most polite sentence?

Answer 1

All are possible, and all are basically asking the same question.

Yes, #9 along with #s 5 and 7 are the most formal (polite).