Learn Hardcore Russian: Я интересуюсь культурой и традициями разных стран. - I am interested in the culture and traditions of different countries.

интересуюсь is the 1st person singular present tense of the reflexive imperfective verb интересоваться (“to be interested in”). You take the stem интересу- and add -юсь because the subject is я. The imperfective aspect shows an ongoing or habitual action.

Which case does интересоваться require, and why are культурой and традициями in the instrumental?

The verb интересоваться always takes the instrumental case to mark the object of interest (answers the question чем?).
• интересуюсь чем? – культурой, традициями.

Why is разных стран in the genitive plural, and what role does it play?

разных стран modifies культурой и традициями and indicates “of different countries.” To express belonging or “of,” Russian uses the genitive case:
• культурой и традициями чего? – разных стран.
Both the adjective разных and the noun стран are in genitive plural.

Can you say я интересуюсь о культуре by adding the preposition о?

No. Unlike English (“interested in something”), Russian интересоваться takes the instrumental directly, without о. Saying интересуюсь о культуре is ungrammatical.

How does я интересуюсь культурой… differ from мне интересно культура…?

• Я интересуюсь культурой Японии is an active construction: “I take an interest in Japanese culture.”
• Мне интересна культура Японии is a passive construction: “Japanese culture is interesting to me.”
The first uses интересоваться + instrumental; the second uses интересен/интересна + dative (мне).

Could you replace разных with нескольких or различных, and how would the nuance change?

Yes.
• традициями нескольких стран = “traditions of several (a few) countries” (emphasizes a limited number).
• традициями различных стран = “traditions of various/different countries” (more formal, very close to разных).

How would you ask someone “What are you interested in?” in Russian?

Use the instrumental question word чем and the 2nd person form:
Чем ты интересуешься?