Learn Italian: Puoi chiedere se i loro caratteri sono compatibili. - You can ask if their characters are compatible.

Se translates to "if" or "whether". When you report a yes/no question indirectly (the original direct question being "Are their personalities compatible?"), you bridge the main verb (chiedere) and the question with se, exactly as you do in English.

Why does loro have the article i in front of it?

In Italian, possessives almost always need a definite article. While most possessives change their endings to match the noun, loro (their) is invariable and never changes. The article i is doing all the work to show that the noun it describes (caratteri) is masculine and plural.

Does caratteri mean "characters" like in a movie or a book?

No, in this context caratteri refers to people's personalities, temperaments, or dispositions. If you want to talk about characters in a story, play, or movie, you would use the word personaggi.

I've heard indirect questions sometimes take the subjunctive. Why is sono in the indicative mood here?

While formal or written Italian might sometimes use the subjunctive in indirect questions, everyday spoken Italian frequently uses the regular indicative mood (sono). This is completely natural and correct, especially for straightforward, factual questions in the present tense.