Learn Croatian: Je li dostojna ovog zaposlenja? - Is she worthy of this employment?

The adjective dostojna (worthy/deserving) requires the Genitive case for the object of that worth. This is part of a broader pattern in Croatian where adjectives expressing worth, lack, or awareness take the Genitive, similar to how we use the preposition 'of' in English.

How do we know the subject is 'she' when the word ona is missing?

The adjective dostojna acts as the predicate and is in the feminine singular form. If the implied subject were male, the adjective would take its masculine form, making the question Je li dostojan ovog zaposlenja?

Could we use the adjective vrijedna instead of dostojna to mean 'worthy'?

You could, and vrijedna also correctly takes the Genitive case, but dostojna is a much better fit for this context. Dostojan implies being deserving, respectable, or fit for a specific honor or role. Vrijedan is more often used for tangible value (being worth money) or as a character trait meaning 'hard-working'.

What is the dictionary form of zaposlenja?

The nominative form is zaposlenje (employment), which is a neuter noun. It is derived from the verb zaposliti (to employ) and follows a very common pattern of creating an abstract noun from a verb stem.