Learn Hardcore Japanese: koko de kutu wo nuide mo ii desu ka?

The particle で marks the location where an action takes place. In ここで, it means “at this place.” If you used に instead, it would suggest a destination or existence (“to here” or “at here” in a static sense), not “doing something here.”

Why is 脱いで used instead of 脱ぐ?

脱いで is the て-form of 脱ぐ (to take off). When you want to attach もいいですか to ask permission, you need the verb in its て-form. The structure is:
Verb(て-form) + もいいですか?
So 脱いで + もいいですか gives you “May I take off…?”

What does the pattern ～てもいいですか mean?

This is the standard polite way to ask for permission:
Verb(て-form) + もいいですか？
Literally “Even if I do X, is it good?” → “May I do X?”
In our sentence, 脱いで (take off) + もいいですか = “May I take off (my shoes)?”

Why is there a も before いい?

The も is part of the fixed expression てもいい. Without it, ていい is either ungrammatical in formal speech or a very casual/slangy version. Always remember:
～て + も + いい = permission.
Dropping も is possible only in very casual contexts.

Why does the sentence end with ですか, and can we omit it?

ですか makes the question polite and formal. If you’re speaking casually to friends or family, you can drop ですか and simply say:
ここで靴を脱いでもいい?
or even add a friendly particle:
ここで靴を脱いでもいいよ？

Why is を used with 靴を脱ぐ?

脱ぐ (to take off [clothes, shoes]) is a transitive verb, which takes a direct object. The object here is 靴 (shoes), so you mark it with を:
靴を脱ぐ = “to take off shoes.”

Can we change the word order of this sentence?

Yes. Japanese allows flexible word order as long as particles stay attached. For example, you could say:
靴をここで脱いでもいいですか？
It still means “May I take off my shoes here?” The emphasis just shifts slightly.

How would you answer this question in Japanese to say “Yes, you may” or “No, you may not”?

To grant permission:
はい、いいですよ。 (Yes, it’s fine.)
どうぞ。 (Go ahead.)
To refuse or ask someone not to:
すみません、ここでは脱がないでください。 (Sorry, please don’t take them off here.)
だめです。 (That’s not allowed.)

What’s the difference between ～てもいいですか and just ～てもいい?

～てもいいですか？ is a polite question asking permission (“May I…?”).
～てもいい (without ですか) can be:
• A casual question (“May I…?” among close friends).
• A statement of permission (“You can…”) when you add something like よ: 入ってもいいよ。 (“You may enter.”)