Learn Swedish: Ibland hoppar det ut en liten apa ur buren. - Sometimes a small monkey jumps out of the cage.

This is a classic presentational construction that follows the End-Weight Principle. Because en liten apa is new information (an indefinite subject), Swedish prefers to introduce it later in the sentence. We use the dummy subject det to hold the regular subject spot, allowing the new information to shift towards the end.

Is it grammatically wrong to say Ibland hoppar en liten apa ut ur buren without shifting it?

It is not strictly wrong, but it sounds much less natural. Swedish strongly prefers using this det-construction when a sentence introduces a brand new, indefinite subject with verbs of existence or motion (like arriving, existing, or jumping into the scene).

Where exactly does the delayed subject en liten apa go in the sentence order?

It lands after the verb and its particle. In this sentence, the formal subject det takes the usual subject spot immediately after the verb hoppar. This pushes the real subject en liten apa past the verbal particle ut.

Why does the sentence use both ut and ur? Aren't they saying the same thing?