Learn Finnish: Uskon, että asiantuntija pitää tilannetta uhkana. - I believe that the expert considers the situation a threat.

When the verb pitää is followed by an object in the partitive case (here, tilannetta) and a state/role in the essive case (here, uhkana), it forms a specific structure meaning "to consider [something] as [something]".

Why are the words tilannetta and uhkana in those specific cases?

This is the strict rule for this opinion-expressing structure. The thing being evaluated (tilanne) must be the partitive object (tilannetta). The word describing what it is considered to be (uhka) must take the essive case ending -na/-nä (uhkana).

If I wanted to use luulla (to suppose / mistakenly think) instead of pitää, would the endings change?

Yes! While pitää requires the essive case (-na/-nä), the verb luulla requires the translative case (-ksi) for the role. If the expert mistakenly thought the situation was a threat, the sentence would be: asiantuntija luulee tilannetta uhkaksi.

Is it possible to drop the word että here, just like we can drop "that" in English ("I believe the expert...")?

No. In Finnish, the conjunction että is mandatory when connecting a main clause like uskon to a subordinate clause. You cannot leave it out.