This is a complete misunderstanding of the curse.
It does not end on his birthday. The actual curse is that he will remain a beast unless he can "learn to love another, and earn her love in return by the time the last petal [of the enchanted rose] fell." The rose is said to bloom until his "twenty first year." Only after that, when the rose begins to die, will the petals start falling. The entirety of the movie takes place
after the prince's 21st birthday. This question is even worse when you remember that the stained glass windows in the beginning show the prince before the curse. And he's clearly a young adult.
See above. This is based on the premise that the prince was a boy when he was cursed. That is false.
If Belle touches the rose and knocks off the last petals, the prince is doomed to stay a beast for the rest of his life. That's the entire crux of the spell. Like I said earlier, it's tied to the rose, NOT his age. The importance of the rose, and specifically of its petals not falling, is established explicitly in the opening narration:
If he could learn to love another, and earn her love in return by the time the last petal fell, the spell would be broken. If not, he would be doomed to remain a beast for all time.
With how much screen time is paid to the rose, I don't know how you could miss it;s importance. The dramatic tension of the climax is achieved though cutaways to the last rose petal falling. Maybe one could miss the opening narration and not know the specifics, but it should be clear by the end that the rose is very important.
After being cursed, the Beast never leaves the castle:
Ashamed of his monstrous form, the beast concealed himself inside his castle, with a magic mirror his only window on the outside world.
I don't think it's out of the realm of possibility that nobody in the town had ever met the prince before the curse (he was at least one social class above them), and therefore no one would have noticed when he became a recluse for a decade.
Incidentally, the songwriter, the late, brilliant Howard Ashman (lyricist for this, The Little Mermaid, and half of Aladdin) would have loved this:
- from page 22 of the excellent booklet that accompanies the music box set
The Music behind the Magic.