Learn Turkish: Kötü şartlara rağmen çalışıyoruz. - Despite the bad conditions, we are working.

The word rağmen (despite) is a postposition that always requires the noun before it to take the dative case (the -(y)a / -(y)e ending). Since şartlar (conditions) ends in a consonant and its last vowel is a back vowel, we add -a to make it şartlara.

Could I use "koşul" instead of "şart" here?

Yes! Koşul is the native Turkish equivalent of the Arabic loanword şart. You will very often hear kötü koşullara rağmen, which means exactly the same thing.

If I wanted to say "Despite the bad condition" (singular), what would it look like?

You would drop the plural suffix -lar. The dative ending -a would then attach directly to the root şart, making the phrase kötü şarta rağmen.

Can I use "rağmen" directly after an adjective, like "kötüye rağmen" (despite the bad)?

Usually, rağmen needs to attach to a specific noun or pronoun. While Turkish sometimes allows adjectives to act as nouns, it sounds much more natural here to include the noun you are describing, such as şartlara (conditions) or duruma (situation).