Learn Swedish: Tror du att de stora problemen är kvar? - Do you believe that the big problems remain?

We use tror because you are asking about a belief or a guess regarding a fact. If you used tycker, you would be asking for a subjective opinion, like whether they like the problems. Tänker is only used for the active process of thinking or planning to do something.

Why does the phrase for the big problems have both de at the start and -en at the end of problemen?

This is the Swedish double definite rule! When a definite noun has an adjective attached, you need the standalone article de, the adjective must take its -a ending (stora), and the noun itself must still take its definite suffix (problemen).

Since problem is an ett-word, shouldn't its plural just be problem?

Yes, the indefinite plural (problems) of ett problem is just problem. However, in this sentence we need the definite plural (the problems). For ett-words ending in a consonant, the definite plural suffix is -en, which makes it problemen.

How exactly does är kvar translate to remain?

The word kvar means left or remaining. So är kvar literally translates to are left. In English, it is often more natural to translate this simply as remain.