Because "Sie" is capitalized, this is the formal form of you.
"Mögen sie schwarze Katzen?" = "Do they like black cats?"
"Mögen Sie schwarze Katzen?" = "Do you like black cats?"
You can't. If duo gives you this as a listening exercise, it will accept either "sie" or "Sie" as being correct
This page explains it very well : www.nthuleen.com/teach/grammar/adjektivendungenexpl.html . Read it, keep it at hand, and refer to the flowchart each time you encounter an adjective here, and you will learn these declensions before you realize it!
schwarzen Katzen would be in Dative Case, this is Accusative case. E.g., Ich danke schwarzen Katzen.
I am still confused according to table http://german.about.com/library/weekly/aa030298.htm
it is plural so it should be schwarzen.
If the sentenced used a plural definite or indefinite article (die, keine, meine, alle, etc.) before the adjective, then the adjective automatically uses the -en declension. This Wikipedia article explains it a bit more clearly:
You can use "Sie" almost like "you" as a polite address to everyone. It is the same in the singular and the plural unlike "du" and "ihr" (used in a talk with friends, children and in the internet): Magst du schwarze Katzen? Mögt ihr schwarze Katzen?