Beauty And The Beast Disney Story

Why did the Beast from Disney’s Beauty and the Beast opened his own door?

So the story goes that an old beggar women seeking refuge asked the prince for shelter in exchange she would give him a rose. The selfish prince sneered at the rose and sent her away. She tried to convince him, but the stubborn prince didn’t change his mind. But then the old beggar women turns into an enchantress and turns the prince into a beast due to his unkindness and so on and so forth. In the opening sequence a glass window depicts the prince and his open door with the beggar women outside. But what I don’t understand is why did the prince open his door in the first place? He had many servants, at least one must of had the responsibility of addressing guests, and if the prince didn’t want to see her, he wouldn’t have even made contact with her at all. Plus if he was such a bad prince, like the story goes, then wouldn’t he have her imprisoned or some other kind of punishment for disturbing him? All he did was said no, nothing more then anyone else would have done.

And how OLD was the prince when he did this?
And why must the servents suffer for a pissy monarch?
And what did the DOG to wrong? (He got turned into a footstool)
And what is the prince’s name?

There are a huge amount of problems with this