llevar (without se) most often means to carry, to take, or to wear.
llevarse bien (con alguien) is a reflexive / reciprocal verb meaning to get along (with someone).
In your sentence, se lleva bien expresses that people get along with one another.
se lleva is 3rd person singular, present indicative of llevarse.
Here, the present indicates:
Yes, you can say en clase (without la). Both are correct, but there is a nuance:
In this sentence, en la clase suggests within that particular class group.
Yes, that word order is perfectly natural: En la clase todo el mundo se lleva bien.
Meaning doesn’t change; you just emphasize the context first:
You could also say Todo el mundo en la clase se lleva bien, which is also fine.
Yes, Todos se llevan bien en la clase is correct and very common. Differences:
Todo el mundo
Todos
Meaning is essentially the same here: everyone gets along.
On its own, Todo el mundo se lleva bien en la clase usually implies that the classmates get along with each other.
If someone wanted to be explicit about the teacher, they’d typically say:
Context would clarify whether the teacher is included, but the default reading is “with each other.”