1. Billy Elliot is a 11-year-old boy who lives in a small town in England.

(What does 'in England' modify? Does it modify 'lives' or 'a small town'?)

2. She asks him to take ballet class.
3. She asks him to take a ballet class.
4. She asks him to take the ballet class.

(Are the three all grammatical? Which one is commonly used?)

the TOWN is in England

3 and 4 are fine; 2 is questionable.

3 means any old ballet class.
4 means a specific ballet class.

"In England" modifies town.

3 and 4 are best. A ballet class indicates any ballet class. The ballet class refers to a specific ballet class.

1. In the sentence "Billy Elliot is an 11-year-old boy who lives in a small town in England," the phrase "in England" modifies the noun phrase "a small town." It provides additional information about where the small town is located.

2. "She asks him to take ballet class." This sentence is grammatically correct.

3. "She asks him to take a ballet class." This sentence is also grammatically correct.

4. "She asks him to take the ballet class." This sentence is grammatically correct as well.

All three sentences are grammatically correct, but the most commonly used construction would be the second sentence: "She asks him to take a ballet class."