Learn Swedish: Företagskulturen är viktig. - The company culture is important.

It is a linking "s" (known as a foge-s). When joining two nouns like företag (company) and kultur (culture) to make a single compound word, Swedish often adds an s to glue them together smoothly.

Is the "s" in företagskulturen a possessive "s", meaning "the company's culture"?

Grammatically, no. While it translates beautifully to "the company's culture", the s here is purely a structural link used to build the compound noun. If you wanted to use a true possessive, you would write them as two separate words: företagets kultur.

Företag is an "ett" word, but kultur is an "en" word. Which gender does the combined word take?

A compound word always takes the gender of its last part. Since kultur is an en-word, the whole compound företagskultur behaves as an en-word. This is why the adjective at the end of the sentence is viktig (the en-word form) and not viktigt.

How do we make a compound word definite, like "the" company culture here?

You only add the definite ending to the very last noun in the chain. The first part (företag) stays in its base form, and the final part (kultur) gets the definite ending -en, giving us företagskulturen.