Learn Turkish: Sisten dolayı uçağın havalanacağı yok. - Because of the fog, the plane has no intention of taking off.

This is the core of the -(y)AcAğI yok pattern. You take the future participle -AcAk, add a possessive ending (here, -ı for third person), and follow it with yok. Literally, it means "its future taking-off does not exist." We use it to express that someone has no intention of doing something, or that an event is highly unlikely to happen anytime soon.

Why is the plane marked with the genitive case as uçağın?

Because the nominalized verb havalanacağı acts as a possessed noun. To indicate who or what is doing the action, we use a standard genitive-possessive construction. Therefore, uçağın havalanacağı literally translates to "the plane's taking off."

Why does havalanacak end up spelled with a ğ here?

When you add the future suffix -acak to the verb stem havalan-, you get havalanacak. However, because the third-person possessive suffix -ı starts with a vowel, the final k undergoes consonant mutation and softens into a ğ.

The translation mentions intention. Can we use this pattern for inanimate objects like a plane?

Yes! While for people this pattern translates perfectly to "has no intention of," for inanimate objects (like a plane or the weather) it generally means "it doesn't look like it will [verb]" or "there is no sign of it [verb]ing." Translating it as "has no intention" personifies the plane a bit, which nicely captures the casual, slightly exasperated flavor of this Turkish phrase.