Learn Hardcore Turkish: Yazın sıcak havada serinlemek için dondurma alabilmek büyük keyif. - Being able to buy ice cream to cool off in the summer heat is a great pleasure.

yazın here is not the 3rd-person genitive (“your summer”) but a special seasonal adverbial form of yaz (“summer”) meaning in summer. Turkish often attaches -ın (or -n) to season names (kışın, baharda vs.) to express “during that season.” You could also say yazda (locative -da), but yazın is more idiomatic for “in summer.”

What case is sıcak havada, and why is hava marked like that?

sıcak havada uses the locative case (suffix -da with vowel harmony) on hava to mean in the hot weather. Here sıcak (“hot”) describes hava (“air/weather”), and the locative tells us where or when the “cooling off” happens.

How is serinlemek formed from serin, and what does serinlemek için mean?

serin = adjective “cool”
serin-le- = verb-forming suffix -le-, “to become/get cool”
serinle-mek = infinitive -mek, “to cool off” or “to become cool”
serinlemek için = purpose marker için after an infinitive, meaning “in order to cool off.”

How is alabilmek formed, and what nuance does it add compared to almak?

al- = verb root “take/buy”
-abil- = potential suffix “to be able to”
-mek = infinitive
Putting them together, al-abil-mek means “to be able to get/buy.” While almak simply means “to buy,” alabilmek emphasizes the ability to do it.

Why isn’t there an accusative -ı on dondurma in dondurma alabilmek?

In Turkish, indefinite direct objects within infinitive (or other non-finite) clauses normally stay unmarked. Here dondurma (“ice cream,” indefinite) has no -ı. If you wanted something like “to be able to get the ice cream,” you’d use dondurmayı with the accusative -ı.

What is büyük keyif doing at the end of the sentence, and why is there no verb like “is”?

büyük keyif (“a great pleasure”) is the predicate noun of a nominal sentence whose “subject” is the infinitive phrase yazın sıcak havada serinlemek için dondurma alabilmek. In Turkish, the copula (equivalent of “is”) is omitted in the simple present, so you just state the subject followed by the predicate noun.

How fixed is the word order here? Could we start with büyük keyif or rearrange the phrases?

Turkish word order is fairly flexible, but the neutral pattern for a nominal sentence is:
Subject (the full infinitive clause) + Predicate (büyük keyif).
You can front büyük keyif for emphasis—
Büyük keyif, yazın sıcak havada serinlemek için dondurma alabilmek.
—but that shifts the focus onto “the great pleasure.” Purpose clauses like serinlemek için usually precede the main verb as shown.