Actually, the word for "korean language" is "한글" (Hangul) but 한국 (Hanguk) is really pronounced like this
한글 means the Korean writing system (or Korean letters), but that's not what I'm hearing here. Maybe the recording is a bit off depending on the device you're using?
우리나라 was given as the correct translation here. I was not given "한국" as a choice in the multiple choice question. Is that correct or is this a glitch?
It is correct. 한국 means Korea, 우리나라 is used by Koreans to refer to Korea, literally meaning "our country". The same goes for 한국어 / 한국말 and 우리말, where the first two mean Korean (the first one is more formal, the second one more colloquial) and the second word is used by Koreans to refer to Korean, literally meaning "our language". :)
I had no "한국" answer but had "우리나라" which is "our country" if I'm correct and I thought that might work heh