Beauty & The Beast plot holes or what happens when I don't get enough sleep

In Everything's Eventual Stephen King* says that he does some of his best thinking in the shower. His thinking involves writing bestselling novels. Mine involves giving way too much thought to Disney movies. In this case Beauty and the Beast. I don't really know what sparked this. I didn't see anything about it and I can't remember the last time I watched the movie, but apparently my subconscious has been stewing over this for awhile now and wanted to talk through it. And for whatever reason lately my body has decided at 3pm it needs to sleep and when I don't give in to its nap time requests it punishes me by not sleeping at night. So what I'm saying is I'm over-tired, and there's a good chance nothing that follows makes any sense. 

She's reading. That makes this relevant.

For those unfamiliar, here's a quick recap of the prologue to the movie. Or you can just watch this clip
A young Prince is a selfish jerk. One day an ugly old lady shows up at the castle and asks for someplace to stay for the night, but says the only payment she can offer is a rose. The Prince is all like "ew gross" and says no way. The old lady tells him not to be deceived by appearances and please let her stay. Prince-y still says no. So then the old lady turns out to be a beautiful enchantress (because she couldn't really be powerful and ugly. Come on now) and puts a spell on the Prince to make him a beast and turn all of the staff into teapots and whatnot, because fuck the poor servants. That rose that she gave him was magic and was going to stay alive until the Prince's 21st year. If he can find love and be loved in return before the rose dies, he'll turn back into a person. If not he's a Beast forever. Sad for him. Years pass and Beast gets really depressed and loses hope and figures he'll just be a Beast forever and then the movie begins!

So here are my problems with this:
1. The intro identifies the Beast as a Prince. So that means there's a King and Queen somewhere, right? If there wasn't, Prince would actually be King. Where the hell are his parents? Did they get turned into pots and beds too? And if so, why doesn't anyone mention them?

2. If he is a Prince, what does his family rule over? No one seems too concerned that their royal family is suddenly gone one day and there's this power vacuum.

3. So maybe we can assume that the guy's name is Prince and he's not actually a prince. Just a really rich guy that makes some pretentious but ultimately catchy tunes. Again, why didn't people in the town notice when he disappeared and suddenly there's this beast thing roaming around? I mean, the town is overly concerned with the fact that Belle reads books**. This seems like the type of place where everyone is all up in everyone else's business. You'd think "crazy rich guy suddenly disappears" would be big news.

4. So the deal is Prince/Beast has to find love by his 21st year. In the song "Be Our Guest" Lumiere says "10 years we've been rusting..." 10 years. So Prince was roughly 11 when he acted like a douche to the old lady/enchantress. This seems like a really harsh punishment for an 11 year old.

Some people spend their time thinking important thoughts and fixing actual problems. I poke holes in the logic of a twenty year old family animated movie.

*This post is going to be light on actual book connections. Here's your first one.
** This is your second book reference and how I'm justifying posting this up here. Also it's my blog so I'll post whatever random thoughts I might have. Especially when I share these thoughts with Boyfriend who reacts by shaking his head and walking out of the room.