No, wo nv pengyou is fine

It's optional.

You're right. It's fine without the de in this case

It's ok when talking but better write it out when learning

It's ok when talking but better write it out when learning

Irl they're most interchangeable afaik, but I think 中文 is more often used for written language.

中文 is more about the literature and the written language. 汉语 - language of the Han people, Chinese. But in many cases you can use both.

I think it should be.