Learn French: Nous étudions une caractéristique telle que la culture. - We study a characteristic such as culture.

The phrase tel que (such as) must agree in gender and number with the noun it refers back to, which is the category being given an example of. Here, it refers to une caractéristique, which is feminine singular, so it becomes telle que. Notice that it agrees with the noun before it, not the noun that follows it.

Why do we say la culture with an article when English just says "such as culture"?

In English, we usually drop the article for general concepts. In French, abstract nouns and broad categories almost always require a definite article (le, la, or les). Therefore, general "culture" must be translated as la culture.

Is caractéristique a noun here? I am used to seeing it as an adjective.

Yes, it is a feminine noun in this sentence. Just like in English, where "characteristic" can describe something (an adjective) or be the feature itself (a noun), French uses une caractéristique to mean "a feature" or "a trait".