"Do you guys listen to Chinese songs?"
It should use 中文 instead of 汉语, no one uses 汉语 in conversation, it's a formal term.
It depends whether you are talking about Chinese culture or a specific Chinese language (remember there are at least 4). 中国 is the place. 中文 refers to generic Chinese culture and language. 汉语 is what you are studying here, what (english-speaking) westerners call 'mandarin', which is what most (but not all) Chinese people speak.
To be even more pedantic, this course seems to be more strongly influenced by Beijing/Northern Chinese dialects.
Seems like both versions are commonly used, at least in writing. Google result counts: "中文歌" 127, "汉语歌" 99, "漢語歌" 104.
On 百度, both simplified versions return 76 pages of results, which looks like it might be a hard result limit.