"Der Chef" or "die Chefin" actually mean "the boss" or "the leader" in English exactly like the French word "le chef", whereas "chef" in English is just another term for a cook; usually the head (get it?) cook of a restaurant.
English also has "chief" which also means "boss" or "leader."
All of these words in German, French, and English come from the same Old French word "chief." The etymology actualy gets more interesting than that--such as how it relates to the German origin for "Haupt" and English word "head"--but that's a bit of a tangent.
Anyway, hope this helps!
Does 'Der Koch' mean 'The Cook' generally, or does it's meaning lean to any specific gender (m/f)?
It's technically the male term (female would be die Köchin). There might be some situations where the male term is used as the 'generic' term, but usually you'd try to avoid that. There's a bit of a discussion here: https://en.wikipedia.org/wiki/Gender_neutrality_in_languages_with_grammatical_gender#German