"We are learning Chinese."
You can, but I would say it is more natural to say "我们在学中文" or "我们在学习". It is less common for 学习 to carry a direct object.
Actually in my daily life I never would use the term 汉语, I only use 中文 (e.g. when I say "Please write it down in Chinese"). It is also my impression for the habit of people in Mainland China and Taiwan too, although I can't speak for all of them. I understand that when talking about teaching Chinese as a foreign language, usually 汉语 would be used. IMHO both terms can be well understood.
I think of something additional… the "Chinese language" we usually refer to is actually the language used by the Han race. Other races in China have their own languages and some of them are in other linguistic categories, such as Uyghur and Tibetian. At least by law, all races literally have the right to use and develop their own language, whilst the only common language is Putonghua, which is 汉语 in substance. So if one day I meet with a Tibetan I would say 汉语 instead of 中文.
The original sentence doesn't imply that the action takes place at the current moment thus requires the use of 在.
I don't think the progressive aspect has to be marked in Chinese in this sentence. I don't think 我学汉语 is grammatically wrong and I don't think it implies it happens on a regular basis.