Learn Hardcore Japanese: gaikoku no bunka ha omosiroi to omoimasu. - I think foreign cultures are interesting.

の is the genitive (possessive/attributive) particle. It links two nouns and indicates that the second noun (文化) belongs to or is related to the first noun (外国). So 外国の文化 means “the culture of foreign countries.”

Why is は used after 文化 instead of が?

は is the topic marker. It sets 外国の文化 as the topic of discussion (“As for foreign cultures…”). Using が would instead mark it as the subject and put focus on identifying or emphasizing foreign cultures, whereas は makes it a general statement about them.

Do we need to include 私は at the beginning of the sentence?

No. Japanese often omits the subject when it’s clear from context. Here it’s obvious you are talking about your own thoughts, so 私は (“As for me”) can be left out. If you want to be explicit, you could say 私は外国の文化は面白いと思います。

What is the function of と after 面白い?

と is the quotative particle. It marks the preceding phrase (面白い) as the content of your thought. You can think of it as signaling “that” in English: “…that it’s interesting.” Without と, you can’t directly connect the adjective to 思います in this way.

Why does the sentence end with 思います instead of 思う?

思います is the polite (masu-form) of 思う. Using 思います makes your statement more polite and suitable for most social situations. 思う is the plain/casual form, used with friends or in informal writing.

How would I say this sentence in casual speech?

In casual speech you can use the plain verb form and a colloquial topic marker. For example:
外国の文化って面白いと思う。
Here, って replaces は informally, and 思う is the plain form.

For a な-adjective or noun, do I handle と the same way?

Not exactly. With な-adjectives (and nouns), you usually include the copula だ before と. For example:
静かだと思います。
(“I think that it’s quiet.”)
Dropping だ is nonstandard in formal Japanese.

Could I say 外国文化 without the の?

While 外国文化 can be understood and appears in some compound words, it’s far more natural in everyday speech and writing to include の: 外国の文化. Omitting の tends to sound technical or stilted.

What’s the nuance difference between 外国の文化は面白いです and 外国の文化は面白いと思います?

外国の文化は面白いです states your observation more directly: “Foreign cultures are interesting.”
外国の文化は面白いと思います emphasizes that it’s your personal opinion: “I think that foreign cultures are interesting.” The latter is softer and clearly framed as your viewpoint.