Learn Turkish: Durum umutsuz, bu yağmurun dineceği yok. - The situation is hopeless, this rain has no intention of stopping.

This uses the lesson's target structure for lack of intention: Verb + -(y)AcAK + possessive + yok. Here, din- (to subside) gets the future participle -ecek and the 3rd person possessive -i (causing the k to soften to ğ). Literally, it means 'its future stopping does not exist.' It is a common idiomatic way to say that something shows zero signs of happening.

Why is 'this rain' in the genitive case (bu yağmurun) instead of just being a plain subject?

Because the grammar of this structure revolves around possession. It forms a genitive-possessive phrase where bu yağmurun (this rain's) is the owner, and dineceği (its future stopping) is the possessed object. You are literally saying 'this rain's future stopping doesn't exist.'

Why use the verb dinmek here instead of a standard word for stop, like durmak?

Dinmek specifically means 'to subside,' 'to calm down,' or 'to cease.' In Turkish, it is the most natural and standard verb to use when talking about weather phenomena like rain (yağmur) or wind (rüzgar), as well as things like physical pain or crying.

Can I use this same structure with var to say the rain does have an intention of stopping?