Hey my dear fellows!

maybe we can have a topic just with little knowledge questions. If there's something like this already, let me know!

I'll start:

Can someone tell me the difference between 中华 and 中国, please? Is it like 中华 is more formal? I just don't want to mistake these both anymore.

According to my C-C dictionary, while previously 中华 referred to the Huanghe River basin and birthplace of the Han people, nowadays it's just another way to refer to China. It gives the example 中华儿女, so China's sons and daughters. That sounds pretty formal to me. It's also used in names, like 中华人民共和国 (The People's Republic of China). So it being formal sounds about right. 中国 is the one you should be using most.

I don't know if that's correct, but if you compare 中国人名共和国 and 中国, could it be that 中国 is just the abbreviated form?

I don't think so. I assume 中国 was already in use in the days of the emperors, but 中华人民共和国 would be the People's Republic of China, a relatively recent development. 人民 means the people. Under the current system they even have a currency named after them, the 人民币。 Of course, the concept of a republic, 共和国，has been around longer than Marx and Lenin. I think Plato wrote about it. If you want to see a government by the "people" that is also "democratic" you must go to North Korea, officially known as the Democratic People's Republic of Korea: 朝鲜民主主义人民共和国 literally the Korea ("Chosun") democracy people republic. Better hurry though, in case it goes the way of the German Democratic Republic (former East Germany), the 德意志民主共和国. (Deutsche Democratic Republic).