Unfortunately, there is not a single, complete Chinese etymology dictionary in this world yet. Therefore, it could be very hard for you to even find the etymology for a certain word.
However, you may try your luck if it's a loan word (like 取缔, surprisingly, it's from Japanese) in 汉语外来词词典 (published in 1984).
Yes it's a coincidence. The characters are completely unrelated.
办 ban4 means "to do, to handle", and it's a phono-semantic compound where the 力 part (strength) conveys the meaning, and the two dots at the sides are a simplification of 辛 (xin1). But the phonetic part when doubled is pronounced bian4. Compare the simplified characters 辩 （as in 辩论）...
Just a coincidence.
What you found is similar pronunciation of 上班, 办公 and 工作 in Mandarin, not similar pronunciation of 上班, 办公 and 工作 in Chinese.
If you get this point, you will realize that these three words appeared before Mandarin was created.
The Austronesian hypothesis for the origin of Min Nan bah (as quoted on English Wiktionary, as of June 2020), comes from Deng Xiaohua's 1994 paper 〈南方漢語中的古南島語成分〉 ("Proto-Austronesian in Southern Chinese Languages"). I have attached an image here from a secondary source:
I see that the 16/17th century (Zhangzhou / Philippine) Hokkien-Spanish ...
A common usage of 我去 is 我了个去 or 哎呀我去. It's just used to express surprise，ridiculousness, astonishment, etc..
It's not relevant to "damn, shit, fxxk" or anything. I think it's more like "what the heck/hell" in English.
I also disagree that it's short for 我去你的. 我去你的 is a dismissal phrase while 我去 doesn't connote that sense.
去你的 : (verb)
"我去你的" = "I (am saying this): damn!"
去你的 literally mean "go yours" (implying 'go to hell')
我去 is a proclamation. It is similar to "Damn!/ Shit!/ FXXK!" " in English.
Actually the most similar pronunciation "baby" in Chinese is běi bí 北鼻, 北鼻 is a loan word, we adopted this word from “baby”, so the pronunciation is so similar. 北鼻 is commonly used for calling friends or lovers on the INTERNET or TEXT MESSAGE.
Chinese people use *皇 to describe the dominant person in some areas, 詹皇(LeBron James), but this kind of name can either be adoring or irony, each individual would interpret it subjectively.
So you would see that both supporters and haters all use 川皇.
First, people in China think, Trump does everything in his own way, never listen to others, like a emperor in ancient China.
Second, they think Trump is arrogant, regarded himself as the king of the world.