"The hats are designed by her."
Translation:Τα καπέλα σχεδιάζονται από αυτήν.
Questions like this have come up before... If I remember, this would not be correct as it is the weak form of the pronoun (and actually that is genitive case?). You can have "I see her" (τηv βλέπω) but in this sentence the strong form must be used because "the hat is designed".. by who? "By her" so its like a standalone section.
I'm still learning too so some confirmation would be good :)
Από της is wrong indeed; της is the weak form of the pronoun in the genitive case.
The general rule is that after prepositions you have to use the accusative case, I don't think it it's a 'standalone section' issue although that might help to remember the rule that you cannot use the weak form unless it's 'tied' to a verb.
With regards to using weak pronoun forms before verbs, you can use either the genitive or the accusative, but you need to know more about what objects the verb takes and how to structure the full sentence, i.e. with direct/indirect objects, prepositions before indirect objects etc. For more information on that, please see my comments in discussion here https://forum.duolingo.com/comment/26964167.