"The lady has taken an education."
Translation:Damen har fået sig en uddannelse.
If you translate it literally it means 'The lady has gotten herself an education'.
While technically correct, "damen" and "the lady" are not perfect matches in daily speech. Instead of lady, woman should be used, to translate dame.
Yes. Both in daily life and in Duolingo it is perfectly valid to say "har taget en uddannelse" As to the other possible solution "fået sig en uddannelse", the sentence is correct without "sig", too. But I think there is a nuance to the two solutions - at least to my ears, "tage en uddannelse" focuses on the process, whereas "fået en uddannelse" is more that she got an education at some past point.