Learn Turkish: Havanın gerçekten düzeleceği yok. - The weather really has no intention of improving.

This is a fixed structure using the future participle -(y)AcAk followed by a possessive suffix and the word yok. Literally, it means 'its future-improving does not exist.' We use this pattern to express that someone has no intention of doing something, or that a situation shows absolutely no sign of happening.

Why does the sentence start with Havanın (genitive case) instead of just Hava?

Because the whole clause acts like a possessive noun phrase. Havanın is the 'owner' taking the genitive suffix, and düzeleceği takes the possessive -i suffix. Literally, you are saying 'The weather's improving doesn't exist.'

Since the weather can't have an 'intention,' is this structure normal to use for non-human things?

Yes! While for people it usually translates to 'has no intention of [doing something]', for inanimate objects or situations like the weather, it simply means 'shows no sign of [happening]' or 'there is no likelihood it will [happen]'.

Could I use the verb düzeltmek (düzelteceği) here instead?

No, because düzelmek is intransitive and means 'to get better' or 'to improve' on its own. Düzeltmek is transitive and means 'to fix' or 'to correct [something]'. If you used düzelteceği, it would mean the weather has no intention of fixing something else!