Learn Swedish: Tror du att de vill anställa dig? - Do you think that they want to hire you?

In Swedish, tror means "think" in the sense of believing something or guessing a fact (like whether they will actually hire you). You use tycker to express a personal opinion, and tänker for the active process of thinking in your head.

Why is there no att (to) before anställa (hire)?

Vill (want) acts as a helper (modal) verb here. In Swedish, these verbs are followed directly by the infinitive form of the main verb without an att in between. This is different from English where we say "want to hire", but similar to how we say "must hire" without a "to".

Will de and dig be pronounced exactly as they are spelled?

No, in almost all spoken Swedish, de (they) is pronounced as dom, and dig (you) is pronounced as dej. You will mostly only hear the literal spelling when someone is reading very formally.

Can I drop the word att (that) like we often do in English, as in "Do you think they want to hire you"?

Yes, in casual spoken Swedish you can sometimes drop the linking word att. However, keeping it in is the standard and most natural way to structure the sentence in writing, so it is great practice to include it.