Learn Danish: Rød står for kærlighed.

When talking about a color as a general concept or category name, Danish usually treats it as a noun in its base form, so rød is standard here. However, using the neuter form rødt is also perfectly acceptable and common when referring to "the color red" as an abstract thing.

Could I use betyder instead of står for in this sentence?

You could, but står for is a much better fit. Står for specifically means "stands for" or "represents" symbolically, which is perfect for talking about what colors or symbols signify. Betyder translates directly to "means," which is usually reserved for literal definitions, like translating a word.

Abstract nouns in Danish sometimes take the definite ending, like kærligheden. Why is it just kærlighed here?

While Danish often uses the definite form for abstract concepts in broad, philosophical statements (like kærligheden er blind for "love is blind"), it usually drops the article when the noun acts as a category or the object of a preposition describing what something represents. Since rød is being categorized as a symbol for love, the indefinite, article-less kærlighed is the most natural choice, just like in English.