In Japanese, the particle で seems to have multiple uses:

1. Instrumental: 車{くるま}で大阪{おおさか}に行{い}った。 "I went to Osaka by car."
2. Locative: 図書館{としょかん}で数学{すうがく}を勉強{べんきょう}している。 "I'm studying math in the library."
3. て-form of copula: 水{みず}はきれいで静{しず}かだ。 "The water is pretty and calm."
4. て-form of copula (sentence conjunction): 彼{かれ}は来{く}るので、大丈夫{だいじょうぶ}だ。 "He's coming, so it'll be fine."
5. 連用形{れんようけい} of copula: そうではない。 "That's not how it is."

(It may be possible to analyze #4 as a 連用形, in the 中止形{ちゅうしけい} sense, but I don't think that's what is in modern usage, since there is an implication of temporal order.)

What is the origin of these uses? Do all these でs historically come from one thing?

(I've heard that "で came from にて", but that doesn't really explain much for me in terms of how all these forms came about and if they are really all from the same original thing.)

In Japanese, the particle で seems to have multiple uses:

1. Instrumental: 車{くるま}で大阪{おおさか}に行{い}った。 "I went to Osaka by car."
2. Locative: 図書館{としょかん}で数学{すうがく}を勉強{べんきょう}している。 "I'm studying math in the library."
3. て-form of copula: 水{みず}はきれいで静{しず}かだ。 "The water is pretty and calm."
4. て-form of copula (sentence conjunction): 彼{かれ}は来{く}るので、大丈夫{だいじょうぶ}だ。 "He's coming, so it'll be fine."
5. 連用形{れんようけい} of copula: そうではない。 "That's not how it is."

(It may be possible to analyze #4 as a 連用形, in the 中止形{ちゅうしけい} sense, but I don't think that's what is in modern usage, since there is an implication of temporal order.)

What is the origin of these uses? Do all these でs historically come from one thing?

(I've heard that "で came from にて", but that doesn't really explain much for me in terms of how all these forms came about and if they are really all from the same original thing.)