I agree with that English translation. However, it's really unnatural to say "con mèo rửa mặt của nó/mình", because it can be deduce that "the face" is its face. Possession should only used when it's another's face.
Similar phrase: Brush one's teeth.
I agree with btopps that the better English translation would be "wash its face" instead of "wash the face", it sounds more natural in English. And personally, I think it is not important to translate word by word, I mean "Why does the cat wash its face?" could be a better translation than "Why does the cat wash the face?" for "Vì sao con mèo rửa mặt ?".
"Wash its face" is so set a phrase in English that "wash the face" would most likely mean some face other than its own. Otherwise no native English speaker would say "the face."