Translation:We are learning Chinese.
"we're studying chinese" was rejected and should be accepted I think
Strictly speaking, 漢語 refers to the spoken language and 中文 refers to the written language. But in everyday speech, this nuance isn't very important. I hear 中文 most of the time and I usually use that word when I refer to the Chinese language. 國語 is another popular way to refer to Mandarin Chinese, especially in Hong Kong and Taiwan. 普通話 is the formal way to refer to Mandarin Chinese in mainland China.
在 here is being used to indicate an ongoing condition/operation.
在 is here because you are studying or learning chinese now and continue to learn it. 正在 has the same meaning