"Korea is hot."
I think one is hot in a sense you cant touch, while the other is hot in a way you can touch. So 뜨접습니다 would be used in the sense of 'this tea is hot' because you can touch the tea and feel that it is hot, however with 'korea is hot' you cant touch korea to feel that it is hot. Basically korea has hot to the touch and hot to the feel, but english only has hot in general
I am not quite sure, but 은/는 marks the TOPIC or THEME of a sentence while 이/가 marks the SUBJECT. Could someone confirm this? When I was learning Japanese (not duo) there was a sentence which started (translated) as 일본은 산이 ... which meant “Well about Japan, the mountains are...”, is this the case in Korean too?
Generally speaking, South Koreans use 대한민국 when they're being specific that they're talking about South Korea. If they're talking about Korea, usually they'll say 한국.
But realistically, the reason it wasn't accepted is simply that duolingo put 한국 as the expected answer and have yet to receive any reports that another answer should also be accepted. I bet if you were to click on "report" and select the option for "my answer should be accepted" that 대한민국 would be added.