"He does not speak Chinese."
不会说 implies that he can't speak Chinese, but 不说 can be either he can't speak it or he choose to not speak it voluntarily.
I completely agree with this for a literal translation. It seems weird though, and I would never use a sentence like that in English.
From my understanding 汉语 refers more to the spoken Chinese and 中文 to the written Chinese. This doesn't mean that they are exclusive. In English they are synonymous.
Zhong refers to contemporary where Han teferences trafitional and perhaps north central concentrated historionics.
I wrote 华语 instead of 汉语. Would that be accepted? (I have learnt Chinese before and most of the books back in primary school wrote Chinese as 华语）