Learn Finnish: Kuulimme uhasta tiedustelupalvelun kautta. - We heard about the threat via the intelligence service.

While case endings like the adessive (-lla/-llä) can express the tool used to do something, kautta specifically means 'via', 'through', or 'by means of'. It is highly frequent when expressing an abstract medium or an intermediary channel through which information flows, which is exactly the role of the intelligence service here.

Does kautta always require the genitive case like we see with tiedustelupalvelun?

Yes. The word kautta is a postposition, and the vast majority of Finnish postpositions require the preceding noun to be in the genitive case (-n).

Why is the word for threat (uhka) in the elative case (uhasta)?

The verb kuulla (to hear) requires the elative case (-sta/-stä) when you are talking about hearing about something. If you used the partitive case (uhkaa), it would imply you were physically hearing the noise of the threat itself, rather than receiving information about it.

Where did the 'k' go when uhka became uhasta?