Learn Russian: Деньги играют большую роль в этом вопросе.

In Russian, the word for money, деньги, is a plural-only noun. It has no singular form in modern usage. Therefore, it always takes plural verb forms and plural adjectives, just like the words "scissors" or "glasses" do in English. That is why we use the "they" form играют instead of a singular verb.

What case is большую роль and why doesn't the noun change its ending?

It is in the accusative case because it is the direct object of the verb играют (they play what? -> a big role). Роль is a feminine noun ending in a soft sign. While the adjective changes to the regular feminine accusative большую, the noun роль looks exactly the same as its dictionary form. This is the standard rule for feminine nouns ending in a soft sign in the accusative case.

Does Russian use the exact same idiom "to play a role" as English?

Yes! The phrase играть роль translates directly to "to play a role" and is used in the exact same metaphorical way as in English to mean having an effect or influence on a situation. Just be careful not to mix it up with another very common Russian phrase: иметь значение (to have meaning/significance).

Why is it в этом вопросе instead of в этому вопросе?