Learn Farsi: موتور ماشین زود داغ می‌شود چون کولر کار نمی‌کند. - The car engine gets hot quickly because the air conditioner does not work.

You would use داغ کردن (to make hot) when someone or something is actively heating an object. Because the engine is just getting hot on its own here without a direct object, we use the intransitive شدن (to become) version: داغ شدن. If you wanted to say 'The sun makes the car hot,' you would switch to the transitive کردن version.

The lesson says verbs with کردن are usually transitive. Why does کار نمی‌کند (does not work) not have a direct object?

That is a great exception to notice! While swapping شدن for کردن usually creates a transitive verb pair, کار کردن (to work) is a fixed compound verb that is inherently intransitive. Because کار means 'work,' the verb literally translates to 'doing work,' meaning the noun part of the compound already fills the conceptual object role.

I originally learned زود as 'early' or 'soon.' Is it common to use it for 'quickly' like in this sentence?

Yes! While زود definitely means 'early,' in spoken and everyday Farsi it is also the most common way to say 'quickly' or 'fast' when referring to the speed of a process. You could also use سریع (fast) for 'quickly,' but زود داغ می‌شود sounds very natural and conversational.

Does کولر refer to the car's engine radiator or just the cabin air conditioning?

کولر is specifically the air conditioning system that cools the passengers inside the car (or the AC in a house). If you were talking about the engine's actual cooling system, you would use the word (radiator) instead.