Learn Hardcore Dutch: Zonder ventilator raakt de kamer snel warm.

The preposition zonder (“without”) usually takes a noun in a general or indefinite sense, so you omit the article: zonder ventilator = “without (any) fan.” If you mean a specific fan, you would say zonder de ventilator.

Why is de used before kamer instead of het?

Dutch divides nouns into two genders: common (uses de) and neuter (uses het). kamer is a common-gender noun, so it takes de.

What exactly does raakt … warm mean, and could I use a different verb?

raakt warm literally means “gets warm” or “becomes warm.” A more neutral alternative is wordt warm:
Zonder ventilator wordt de kamer snel warm.
Both are correct; raakt often emphasizes the process of changing state.

Why does the sentence start with Zonder ventilator, and how does that affect the word order?

Dutch follows the verb-second (V2) rule: in a main clause the finite verb must be the second element. By fronting the prepositional phrase Zonder ventilator, you push raakt into second position, and the subject de kamer comes after the verb.

Why is snel placed before warm, and can I move it elsewhere?

snel is an adverb modifying the adjective warm, so it directly precedes it: snel warm. You can shift it for emphasis, but the most natural spots are right before the adjective or at the very beginning of the sentence:

Snel raakt de kamer zonder ventilator warm. (emphasizes how quickly)
Zonder ventilator raakt de kamer warm, snel. (less common and more colloquial)

Could you also say Zonder ventilator wordt het snel warm in de kamer?

Yes. That version uses the “dummy” pronoun het (like English “it” in “it’s getting warm”) and places in de kamer at the end. It’s perfectly natural, though it shifts the focus slightly to the fact that “it” (the temperature) rises quickly.