"I live in a city called Kocaeli which borders Istanbul." *You can use "borders" as a verb to mean that something (e.g city) forms or grows along the edge of something else. *It is alright for you to use "a city called" as Kocaeli may not be as well-known as Istanbul to some readers. If you still wish to use "border" as a noun, you can write: "I live in the city called Kocaeli which [shares] a border [with] Istanbul." *You use "with" to express accompaniment.

"I live in the city called Kocaeli which has a border to Istanbul." That's perfectly correct, but in conversation you would more often drop "called" and just say: "I live in the city Kocaeli which has a border to Istanbul".

I agree with the verb form and "shares a border with", I've never heard "the border to". Though I've never heard anyone use border talking of cities, border wouldn't be my preference at all. My city is on the outskirts of Istanbul.

I would say "I live in a city called Kocaeli, which borders Istanbul" i.e. The United States borders Mexico The United States shares a border with Mexico Kocaeli is on the border of Istanbul