Learn Finnish: Lasinpuhalluksenko jälkeen he jäähdyttävät sen heti? - Is it after the glassblowing that they cool it down immediately?

This is the core of contrastive focus fronting! By pulling lasinpuhalluksen to the front of the sentence and adding -ko, the speaker is specifically verifying the timing: "Is it after the glassblowing (and not some other step) that they cool it?" Normally, yes/no questions start with the verb, but here the true focus of the question is the glassblowing.

Could the -ko clitic be attached to the postposition jälkeen instead?

Yes, you could absolutely say Lasinpuhalluksen jälkeenkö he jäähdyttävät sen heti? Both are correct. Attaching -ko to the noun (lasinpuhalluksenko) hyper-focuses on the glassblowing itself, whereas attaching it to the postposition focuses on the whole phrase "after the glassblowing".

The verb is jäähdyttävät. How does this differ from the similar verb jäähtyä?

This is a great example of Finnish transitive vs. intransitive verb pairs. Jäähdyttää is transitive, meaning someone does it to something else ("to cool something down"), which is why we have the object sen (it). Jäähtyä is intransitive and means something cools down on its own, for example: Lasi jäähtyy (The glass cools down).

Why is the object sen in the genitive-looking accusative case instead of the partitive?