Learn Turkish: Herkes şimşek çakacağını sanıyor. - Everyone thinks that lightning will flash.

Sure! It comes from çakmak (to flash or strike). We add -acağ (the future clause marker -acak, with the k softened to ğ), then -ı (3rd person singular possessive, linking it to the lightning), then the buffer -n-, and finally the accusative case -ı because the whole phrase 'that lightning will flash' is the direct object of the verb sanıyor.

Usually the subject of an -ecek or -acak clause takes the genitive case. Why is it just şimşek here and not şimşeğin?

You are right that specific subjects of these clauses take the genitive (like benim gideceğimi for 'that I will go'). However, when the subject is a general, non-specific concept or a natural phenomenon like lightning, rain, or snow, Turkish often drops the genitive case. So we say şimşek çakacağını instead of şimşeğin çakacağını.

Why does the sentence use sanıyor instead of düşünüyor for 'thinks'?

Sanmak means to suppose, assume, or guess that something is true. It is the perfect verb here because people are making a prediction or assumption about what the weather will do. Düşünmek is used more for the active, mental process of pondering or having a philosophical opinion.