Learn Farsi: آنها فرهنگ کشوری دیگر را دوست دارند. - They like the culture of another country.

To say "another [noun]", we often add the indefinite suffix -ی (meaning "a" or "an") to the noun, and then put دیگر (other) right after it. Here, کشور (country) becomes کشوری (a country), followed by دیگر. Literally, it means "a country (that is) other".

Is there an ezafe linking فرهنگ (culture) and کشوری (a country)?

Yes! To say "the culture of a country", we use the invisible ezafe vowel (an "e" sound) to connect them. When spoken, it is pronounced فرهنگِ کشوری (farhang-e keshvari).

If کشوری (a country) is indefinite, why do we use the definite object marker را (rā)?

The marker را shows that the direct object of the verb is specific. While the country itself is indefinite ("another country"), the actual thing they like is "the culture". Because they like a specific culture (the culture of that country), the whole phrase is treated as specific and needs را.

Why is را placed after دیگر instead of right after فرهنگ (culture)?

In Farsi, را always goes at the very end of the complete noun phrase it marks. The entire direct object here is فرهنگ کشوری دیگر (the culture of another country), so را waits until the very end of that whole group of words before it appears.