Question: What place was the Beast a Prince of? Considering that Maurice had to inform the other villagers about a castle in the woods. The village is not extremely far from the castle, apparently. Belle gets there soon enough when she wants to return, after helping her father. Were the villagers not ruled over by the Beast's parents?
Answer: It's never specified that the Beast's family were rulers. Until 1789, France was ruled by a monarch. Lower-ranking royals can have titles and live in castles, but it does not mean they hold any political power or position. Some royals or aristocrats may act as an appointed regional governor under the monarch or hold a ceremonial position. For example, Prince William of England is also the Prince of Wales and Prince Harry is the Duke of Sussex, though neither "rule" those districts. Also, this is a fairytale, so details are deliberately vague and generalized.
Question: Gaston tells Belle that she will end up as a beggar if she does not get married. I know this is a fantasy setting, but in similar settings, couldn't an unmarried woman still inherit property?
Answer: Gaston is trying to intimidate her into marrying him. If she did inherit father's home someday, it's likely that she could sell the property and go to Paris or another place with a larger population. Women could work a variety of low-paying jobs. Still not an ideal life, but better than growing old in the village and having very limited options.
Answer: One possibility is that this property belonged to the Prince. This could explain why the witch asked him for shelter, instead of asking a parent.