So if I understand this correctly, 複合語 contain at least one nominalized verb or some portion of a verb, while 熟語 don't contain any, right? So a word like 雨戸 is a 熟語 even though it's based on kun'yomi?
Not really. "Noun + noun" can be 複合語, for instance 海風, 松葉, 砂浜 or 紙コップ. These words are just made from mixture of each component, e.g., 海 "sea" + 風 "wind" = sea wind/sea breeze. So, 雨戸 is 複合語. (Note that 複合語 can be categorized into 熟語 in a broader sense, so it's not wrong to call those words 熟語.)
How is this not 国 を 営んでいる though?
国 is the object of 営む in that case, so the word order should be 営国, as same as 営業, 営利 or the examples in #3.
Also, I love how it's 終始 instead of 始終 as if it's going back in time mind blown
Actually, the word 始終, reading しじゅう, exists. Incidentally, these two words are not the same in meaning. For instance, 彼は終始食べていた。 means that he was eating something from the beginning to the end of a period of time, i.e., he kept on eating throughout a period of time. On the other hand, 彼は始終食べていた。 is a past habit; "he used to always/frequently eat something". Thus, 終始 vs. 始終 is close to ずっと vs. いつも.
Also, 終始 can be the stem of a -suru verb 終始する, meaning "to do in the same state/way/attitude" or "to stick to one's faith", whereas 始終 doesn't have this usage. (As a side note, 左右 vs. 右左 has a similar difference. They are the same in meaning as a noun, but 左右する means "to affect/influence", e.g., 入試は人生を左右する "The entrance exam strongly influences one's life". 右左する doesn't exist.)
Ok I'm not sure I've got this right... were these 熟語 created in China and then imported to Japan, or were they created in Japan based on Japanese sentences (火が出る) but with the Chinese word order in mind?
Also, allow me to ask a broader question at this point: generally speaking, were 熟語 imported from China, or were they created in Japan based on Japanese language and culture using kanji as building blocks?
Both exist. Furthermore, there are words that were made in Japan and exported to China, for instance western origin concepts that were translated in the end of Edo and the beginning of
Meiji Period, e.g., 文化, 文明, 経済, 宗教, 哲学, etc.
they are part of Chinese culture, like 元気 or 風水
Actually, 元気 is a Japanese word. 元気 was originally written as 減気, meaning "illness decreasing and getting better". It changed to 験気 later, and fainally 元気 is used now. 元気 exists also in Chinese, but it has a different meaning from the Japanese one, and vitality/health is written in a different way in Chinese.
Similarly, there are words whose meanings are different in each language, e.g., 手紙 (letter in Japanese, toilet paper in Chinse) or 娘 (daughter in Japanese, mother in Chinse).
Kanji is not so simple, right?