"Where is your mother?"
They are functionally equivalent in these situations, because どこ makes it obvious that you're talking about a location. If I had to say, I guess どこですか is more general and hence more conversational, whereas どこにいますか is probably more "proper" and literary, but the difference is largely negligible on the whole.
The function of に is to indicate the target location of the verb. So, in this case, since the verb is います which means "to exist", どこにいますか means something along the lines of "at/in where does [your mother] exist?"
I might be going more in depth than necessary, but note that the target location (に) is a slightly different concept from the action location (で). The target location describes the intended result of the action, while the action location describes where the action takes place. For example: (bad joke incoming)
- ボールをホームに投げます。"I throw the ball to home plate." From this sentence, you know the ball goes to home plate, but you don't necessarily know where I am.
- ボールをホームで投げます。"On the platform, I throw the ball." From this sentence, you know I'm on the platform, but you don't necessarily know where the ball is going.
- (bad joke explanation: ホーム can mean "home plate" in a baseball context, or it can mean the "platform" where one waits for a train, probably shortened from プラットホーム)
います (and also あります) is kind of an exception in this regard. It's considered a "state" verb rather than an "action" verb, so rather than describing the "act" of existing, it describes the resultant state of some action, e.g. "I walked the store, and now I AM here". That's why you would only use に with います, because it doesn't really describe an action. (By the way, the verb which describes the "action" of existing is 存在する【そんざいする】)
「君のお母さんはどこですか？」didn't go through (nor any noun for second person for that matter). I know that using second person should be avoided in Japanese, but the provided translation could be retranslated back as "Where's mother?" or "Where is OUR mother". Did I do any grammatical mistake? Or does it simply reject explicit possessor? If the latter is the case, shouldn't the actual sentence be changed or the above translation allowed?
You're right; 君 and other second person pronouns may not be commonly used in Japan, but they shouldn't penalize you for using them correctly, as you have here.
That said, it doesn't make your suggestion any more, or less, "correct" than the provided translation. The fact that it can be re-translated back into other English sentences is irrelevant; it's only possible because we have no context here, whereas normal Japanese almost always does.
They are functionally equivalent in this situation, because どこ makes it obvious that you're talking about a location. If I had to say, I guess ですか is more general and hence more conversational, whereas にいますか is probably more "proper" and literary, but the difference is largely negligible on the whole.
For these learning exercises, I personally think it should not be acceptable without か. However, it can be dropped in speech, and commonly is in casual speech, because it can be heavily implied by a questioning tone of voice.
This phenomenon is independent of the fact that the sentence has a question word (どこ) already in it. The か can be dropped from direct yes/no questions too, e.g. スプーンは要ります？【すぷーんはいります？】With the right tone, it would mean "Do you need a spoon?"
Umm, dore means "which". You want どこdoko.
Also, using どこだ？ with an upward inflection is indeed informal, but it is also somewhat confrontational. Usually, おかあさんはどこ？ or おかあさんはどこなの？ (both with upward inflections at the end) would be used in casual situations, with the latter being softer in tone.
There are a few general rules, which talk about a kanji's on'yomi and kun'yomi, but personally, I feel like there are so many exceptions that it's more worthwhile to just rote memorize vocabulary words and get a feel for the "rules" that way, than it is to learn the rules and practice applying them.
The main general rule I'm aware of is that a kanji is pronounced using its on'yomi if it is together with other kanji, but its kun'yomi is used if it is on its own or together with hiragana/katakana (also known as okurigana).
What is the purpose of "にいます" in this sentence? I left out "に" and was marked wrong for it.