Not a native speaker, but how I refer to my Aunts/Uncles in Chinese is by referring to them in birth order in their family. E.g., "1st Aunt" is the first sister born in my mom's family. IIRC, her husband gets a similar moniker (i.e., Husband of 1st Aunt).

But what if your father had a sister who was older than your mother's oldest sister?

Or does Chinese use different words for "mother's sister" and "father's sister"?

Yes Chinese use different words for the two sides of the family.

Also, depending on which part of China you're from, first cousins whose fathers are brothers (thus having the same surnames) would refer to each other as brothers and sisters.

That part is rather confusing sometimes but otherwise Chinese is very precise in enumerating your relationships.

This is quite similar to the way things work in India as well (or at least the parts of Northern India I'm familiar with).

In several dialects there is also a variation of the word for oldest and youngest aunt/uncle/cousin/child/etc.

The middle ones just get referred to by number.

There's also the whole Confucian family numbers to. Sort of a degrees of separation metric in a way.

Wow, by that reckoning I'd have 23 siblings!