"A dog runs faster than a person."
Translation:Ein Hund läuft schneller als eine Person.
What case is "a person" in in this question?
English doesn't distinguish cases with nouns, so that's not really important. And English comparison with "than" is... complicated (with some treating "than" as a conjunction and some as a preposition, so you may hear both "he is faster than I" or "he is faster than me").
In German, the noun after als is always in the same case as the thing you're comparing it with.
Here, you're comparing to the subject ein Hund, so it's als ein Mensch (nominative case).
An example where both nominative and accusative are possible, with different meanings:
- Ich liebe dich mehr als mein Vater.
- Ich liebe dich mehr als meinen Vater.
They're both "I love you more than my father" in English, but in German you can tell the difference between "I love you more than my father loves you" (comparing subjects: mein Vater is nominative like ich) and "I love you more than I love my father" (comparing objects: meinen Vater is accusative like dich).
why person is a woman?
"person" doesn't mean "woman". A person can be a man or a woman, a boy or a girl -- any human regardless of age or gender.
The noun die Person has feminine grammatical gender.
But grammatical gender is mostly irrelevant. Just because die Person is grammatically feminine doesn't mean that a person is female. Kind of like how the English word "long" is not long.
It DOESN'T accept Person AND/OR Mensch!
No, of course not. You have to translate the entire sentence, not just one of its words.
So you have to translate "A dog runs faster than a person" into something along the lines of Ein Hund rennt schneller als eine Person or Ein Hund läuft schneller als ein Mensch. Not just into Person or Mensch -- let alone Person AND Mensch! Person Mensch is definitely not a correct translation.