Continuity Errors - Beauty And The Beast

Beaty And The Beast (animated) is one of my favorite Disney movies, but it is not without fault. I haven't seen the movie in several years, but here are some thoughts/questions the movie fails to address.

• Why did the prince, a child, answer the door for a stranger?
  • Why did a prince answer the door at all?
• The punishment did not match the "crime", especially for a child.
• Why was the entire castle punished for the prince's misdeed?
• Where were the adults/guards when the witch is casting spells on the prince?
• The castle seemed to be within a day or two journey of the town. How had the townspeople never hear about it or the curse?
• After the castle is cursed, is that region no longer ruled? It seems to me that everyone else continued to get along just fine in spite of living in anarchy.