Translation:How is your father?
Because お父さん usually refers to someone else's father, not yours.
By translating it as, "how is father?" you're implying that this father you're asking about is your father too.
I've heard people talking about their own father/mother as お父さん/お母さん and 父さん/母さん in FMA (if I remember correctly) and some Japanese people though.
Either this one is wrong or all these questions in the same lesson are wrong:
Question: お父さんはどこですか？ Correct response: Where is father
Question: お母さんはどこですか？ Correct response: Where is mother
Question: おかえりなさい、お父さん。 Correct response: Welcome back, father
Question: お母さん、行ってきます。 Correct response: Mother I'm leaving.
Question: お母さん、ただいま。 Correct response: Mother I'm home.
It's... not that simple.
First of all, your ultimatum doesn't really hold up because, of the five example sentences you gave, only the first two follow the same usage of お父さん as this question, namely お父さん is the topic of the sentence, not the listener/addressee. In Japanese, who you talk TO (listener), who you talk ABOUT (topic), and how they relate to each other and/or you (speaker) has a significant impact on the kind of language you use.
On the flip side of that, you can learn a lot about people's relationships just from the way they talk to each other. (Of course, this is at least a hundred times more important in business Japanese, but it's still a huge part of everyday situations too.)
As for the correct usage of お父さん (and お母さん), @xyh4l84 is completely right. It is generally used when talking TO someone ABOUT their father (who is also not your father), so the use of お父さん here implies that relationship between speaker, listener, and topic, which is why OP's answer wasn't accepted.
The other word for "father" we've learned so far, 父 (ちち), is generally used when talking TO someone ABOUT your own father. (Same with 母 (はは) for "mother".)
On the other hand, when talking TO your own father ABOUT someone/something, it obviously varies by household and one's specific relationship with one's father, but お父さん is generally used, which is why the first three of your examples are correct (and different from the first two).
To further complicate matters, as @xyh4l84 mentioned, Japanese people sometimes use お父さん when talking TO someone ABOUT their own father, when they really should be using 父. I think the main reason for this is that Japanese children are taught (from a very young age, obviously) that their own father is 「お父さん」. That's what they call their father, that's what their mother will use to talk about their father, sometimes even their father will talk about himself (to the child) using お父さん. It becomes a habit, well before they have any need to refer to their father as 父.
Hence, I think "how is (my) father?" is an appropriate translation in some contexts, but it's not the correct answer.
Yes, that is possible too. However, typically Japanese people don't use formal language (おげんき and です) between family; it is entirely conceivable that the speaker is speaking to their estranged father (hence the formality/distance), but I think that would not be the general interpretation of this sentence.
I made the same mistake, but after reading the explanation above, I tend to disagree with your opinion. The reason being that if you would be trying to imply that it is your own father, you would not use お父さん(otosan), but rather 父 (chichi). It is not even pronounced the same.
Although duolingo did not explain to me why it is not pronounced the same Eben if it IS the same Kanji. I would love an explanation about that.
Just to clarify, Duo is trying to teach us the "technically correct" way to differentiate between お父さん (otousan) and 父 (chichi).
お父さん is to be used when talking ABOUT someone else's father, or when talking TO your own father.
父 is to be used when talking ABOUT your own father.
If you're talking to your siblings about your own father (presumably also their father), the distiction kind of breaks down because families generally have their own names for each other that everyone is familiar with. Also, generally speaking, you wouldn't need to use such polite language with your siblings.
I also chose "father" without "your" because both my husband (Japanese) and I always used "おとうさn" whenever either one of us would address or speak about my husband's father when we stayed with his parents in Japan. I never heard "chi chi" being used except in language lessons.
You also have to remember that while it could be construed as asking about your own father, generally in a question the topic (は) is inferred as the listener. How often do you ask someone, "How is (my) dad?" You of all people should already know that. The only common exception I can think of is asking your mother or a sibling, in which case the language wouldn't be this formal.
The お is an honorific prefix, applied to 元気＝げんき because this word, meaning "health", refers to the other person's father. It would be perfectly correct grammatically to omit the お but it might be less polite.
I am not a native speaker so I don't really know whether it is "okay" or common to omit the お here. I'd be grateful if any native speakers could chime in here...is it okay or common to omit the お or would that be seen as more casual / rude?
First, even if it is very informal, if someone asks, "Is your father good?" it should be assumed that they mean, "Is your father well?" unless... like, your father might actually be an opposing force of some kind. (and unless you are a native speaker in a really odd situation, I do not see this happening)
Second, "Is your father doing well?" is also counted wrong.
In general, people use the question "元気ですか？" to ask "How are/is you/(someone else)?"
Think about it...in English we would rarely ask: "Is your father healthy?" It's not really a natural translation; in English it might come across as socially awkward or even rude to phrase it that way. The Japanese phrase is a common / polite phrase, so a more natural translation of it is "How is your father?"
Just adding to what @cazort said, "healthy" in your sentence would be interpreted as "not sickly" (as an Australian native speaker), so to convey that sentiment in Japanese, you would need to use 健康 (けんこう).
健康 means "well" as in "not sick", whereas 元気 means "well" as in "in good spirits".
Yeah, but it's also rejecting "Is your father well?" which doesn't mean successful, so I don't think it's applying that logic in its rejection.
Plus, also to be pedantic, just because in English "well" has a separate implication doesn't cancel out the fact that "well" also does refers to wellness or a state of being. By that argument, "How is your father?" could also be written off, because you can speak that way in English to gently refer to past events--like if the listener's father had to leave his job or declare bankruptcy, you could be asking "How is your father?" to check if he's more financially stable.
There are many, many ways to say it, even if you're not considering how you can ask indirectly.
Mainly it boils down to the nature of your relationship with your father, how much of that is known (or you want to be known) be the doctor, and your own relationship with the doctor. Obviously, this affects how formal your sentence will be, but also determines whether you use (お)父さん (おとうさん), 父 (ちち), 親父 (おやじ), パパ, or simply [surname]さん to refer to your own father.
Also, it depends on whether you're asking out of general curiosity or genuine concern that something may be wrong. In the former case, you would generally use 元気 (げんき), without the お, while the latter case is more likely to be 大丈夫 (だいじょうぶ) or 無事 (ぶじ) instead. You could also simply say どう instead.
To me, "is your father alright" carries the implication that he was not alright before or there is some reason to believe that he might not be alright. The Japanese sentence doesn't have that same implication; you would use 大丈夫【だいじょうぶ】instead of お元気 if you wanted to get that meaning.