Couldn't this sentence also be translated as;
"Has she taken her the backpack?"
because "jej" also has the meaning of "to her" - ie. dative of "ona"
Or does the "jej" need to come before the verb for that construct (like Spanish for example)?
Yes, it could.
Shouldn't it be swój plecak?
Well, it can be "jej plecak" if Susan took Anna's backpack.
And "swój plecak" if Susan took her own backpack.
Is the word "jej" necessary in that is implied by the subject pronoun "ona"?
Czy ona wzięła plecak? - Did she take the backpack? (Most likely her own, but not necessarily)
Czy ona wzięła swój plecak? - Did she take her backpack? (Her own)
Czy ona wzięła jej plecak? - Did she take her backpack? (Not her own, but someone else's)
Excellent answer. Thank you.
I removed this sentence as using "jej" is not very likely (although technically correct) and created "Czy ona wzięła swój plecak?" instead.