On it's own it's an interjection like the ones you said, but maybe stronger, like "F*** it!", and it can also be used like "f***ing". Literally, it means "his mother's", but might come from the phrase "f*** his mother's c***".

I think it's quite universal. Noone likes other people saying stuff about your mother. In Dutch we have the phrase "je moeder" (你妈) which works the same. I think in most languages the "fuck" is implied. Tu madre. Deine Mutti! I guess one's mother is probably most sacred, especially to guys. In Dutch there are very few jokes about fathers. The only one I know which is quite offensive is: "Go tell your dad to wash his dick because your breath smells." (Implying you performed oral sex on your father.) (你快告诉你爸洗澡他的小弟弟因为你的口臭很重。)

because mom is a very scared word in Chinese life, none of them could permit anyone to abuse the word, it sounds like you are insult their mother.

他妈的means shit or damn. 妈的 and 该死 have the same meaning with 他妈的。

ofc, 他妈的 is the third person. that's all. we use 你妈的 to the person who is listening to us. and we use the third person(他) when there is nobody listening us. anyway, they all come from 操你妈的逼