Learn Russian: Директор очень доброжелательный и благосклонный по отношению к студентам. - The director is very friendly and favorable towards the students.

Both are grammatically correct, but по отношению к (in relation to / towards) sounds a bit more formal and emphasizes the director's specific stance or policy toward the students. Using just к (доброжелательный к студентам) is also fine, but slightly more casual.

The word доброжелательный looks like a compound word. What are its roots?

Spot on! It is made of добро (good) and the root of желать (to wish), plus the adjective suffix. Literally, it means 'wishing good,' which translates to 'friendly' or 'well-disposed.'

Could we use the short-form adjectives (доброжелателен and благосклонен) here instead?

Yes, absolutely! Short-form adjectives are very common when describing someone's behavior towards others. Using the long forms (доброжелательный, благосклонный) as seen in this sentence emphasizes that these are his permanent character traits rather than just a temporary attitude.

What dictates the case of студентам here?

The compound preposition по отношению к ends with the preposition к (to/towards). In Russian, к always dictates the dative case, changing студенты (nominative plural) to студентам.