I have memorized the declension of definite articles--in the sequence of Masculine, Feminine, Neuter, Plural--as:
des der des der
for the Genitive, which I can sing-song as the repetitive "desder desder".
Similarly, for Nominative:
derdie dasdie (der die das die)
and for Akkusativ:
dendie dasdie (den die das die)
(Nom & Akk sind einfach, ja? Oder nur sehr bekannt?)
And for Dative:
demder demden (dem der dem den)
(In all cases, one must use other clues in the sentence/phrase to fully distinguish masculine/neuter or feminine/plural.)
As an added bonus, once I learned these, I could use them as the foundation for remembering/figuring out the indefinite articles (ein eine ein, etc) which gives me just a little more difficulty because there's no unique form for masculine nouns and no form at all for plural nouns.
My abbreviated version is:
R E S E
N E S E
M R M N
S R S R
Which I recite in my head as something sounding like:
"ree-seh, nee-seh, mrr-m'n, srr-srr"
The columns from left-to-right are masculine, feminine, neuter and plural. The rows from top-to-bottom are nominative, accusative, dative and genitive.
Then the next time you get this challenge, you should answer with "The dog of the woman is old" and then flag/report it as "my answer should be accepted".
But you asked why it's not, and the reason it is not is that it is an unnatural way of putting it. The course maintainers likely set up the most common answers and didn't try to add in every awkward, unusual, rare, archaic, and poetic turns of phrase, relying instead on the report feature to identify for consideration any answers they didn't initially adopt.
But they don't read through the comments looking for answers to add. That's what the report feature is for.
Technically perhaps, but in my experience, nobody says it like that, so you may not be understood and you might as well call that "wrong".
Similarly with der Frau Hund which sounds even more wrong to me even though it might technically be correct. (meines Vaters Kind or des Präsidenten Sohn sound a bit better to me, but are still unusual.)
The genitive goes after the noun essentially always in today's German (and the owned noun has an article of its own) -- der Hund der Frau would be the usual way to say this if you're going to use the genitive case.
No, the woman is not male in this sentence.
It's just that all forms of the definite article are ambiguous, so der can be masculine nominative singular, feminine genitive singular, feminine dative singular, or genitive plural, for example.
So if you see der, it's only masculine if it's nominative case -- if it's dative case, it must be feminine, and if it's genitive case, it must be either feminine or plural.
(Here, it's feminine genitive singular.)