"Their book is already here."
Translation:Их книга уже здесь.
Here их=their, a possessive isn't used as них/неё/него. их=них=them.
TL;DR: Н is added:
- after simple prepositions (без, в, для, до, за, из, к, на, над, о, от, по, под, перед, при, про, с, у, через) - через него, до них, над ними
- after adverbial prepositions (возле, вокруг, впереди, мимо, напротив, около, после, посреди, сзади) - возле неё, после него
There are more cases when it's not added but when you'll need it then you can easily read the given link.
"Их", is the owner and the genitive case. "книги", would be the solution for the object, if the sentence would be negated.
своё is only used when you're referring back to the subject of the sentence and it can be used for any type of subject.
This order is not incorrect (There is virtually no such a thing in Russian - the correct order). It's just highly unusual to put these words in this order.
For the same reason "Already here their book is" would be wrong in English. The words are correct but in the wrong order.