"How is the family going?" is common usage where I live in Australia. "How is the family doing?" is much less common. It tends to be an Americanism that we know because of television. Anyone who has visited Australia would be familiar with "How are you going, mate?" as a greeting. Don't misunderstand me. I love Duolingo - I wouldn't still be using it nearly six months on otherwise - but being marked incorrect because I use a non-US but standard translation does get irritating. I make enough mistakes without adding extras.
It's not grammatically wrong at all; this is what we call in linguistics the difference between formal agreement and notional agreement. American English makes formal agreement the standard in most cases (where verbs & adjectives agree with the grammatical form of the noun), while British and Commonwealth English tend to make notional agreement the standard (where verbs & adjectives agree with the notion implicit in the noun).
There is another consideration.
The government has announced new tax regulations.
The government have different views on this matter.
The family is united against the council's decision.
Most of the family are here, but some are still en route.
When considering people collectively one would perhaps choose the singular, but in British English there is the option to use the plural, usually where individual members can be discerned in the meaning. Many Brits use the plural with these collectives in either situation, but I imagine it's a matter of taste and style.
In Ireland we'd typically say "grand" if everyone is in the best mood of their entire life, or more likely something along the lines of
"ah, sure isn't it terrible, Sean is after going back on the drink and..." insert more misery...
In short, it is not unusual to get a negative sounding response.
I agree with all the points on Australian English (being one I also get this wrong constantly) but another point: doesn't 'va' mean 'go' as in 'comment ca va' or 'how it goes', so it makes sense to say 'goes' even if 'doing' is a colloquialism (and thus a more sensical translation) maybe the Australian way is more logical...just saying :P