Learn Russian: Сколько лет, сколько зим! Как ты живёшь?

Literally, this idiom translates to 'How many summers, how many winters!' In Russian, the word for summer (лето) historically also meant 'year.' This is why лет is still used as the genitive plural for years when counting (such as пять лет). The phrase emphasizes the long passage of time by referencing the changing of the seasons.

Why are лет and зим in this specific grammatical case?

They are in the genitive plural case because they follow the word сколько (how many). The base forms are лето (summer/year) and зима (winter), which become лет and зим when pluralized and put into the genitive case after a quantity word.

How does Как ты живёшь? differ from the standard greeting Как дела??

While Как дела? (How are things?) is a great all-purpose greeting, Как ты живёшь? (literally 'How are you living?') is more personal. It asks about the person's life in a broader sense and is typically used when you haven't seen someone in a while, making it the perfect follow-up to this idiom.

Is Сколько лет, сколько зим! appropriate for formal situations?

No, this is an affectionate and colloquial phrase. You should generally use it with friends, family, or acquaintances you haven't seen in a long time. In a strict formal or business setting, it would sound too familiar.