About 2)
a)
I thought you understood 急用ができて correctly, and "had to finish urgent business" is a free translation of it. Then you don't understand できる, either.
Well, it's not that easy for me to be sure wether such a free translation is correct or not ^^
in this case, できる means "to be completed", for example:
急用ができる => "urgent business is being completed/finished." In a more free translation, I could imagine: "I finish urgent business."
For the sentence 急用ができて帰った I would literally translate:
"Urgent business being finished, I returned" or "Because urgent business was being finished, I returned."
Especially the second version is very close to a free translation like "Because I had to finish urgent business, I returned".
But you once told me that -て form is basically always "just" a connector, if I understood it correctly. So for a free translation, if I wanted to be absolutely sure that I don't overinterprete the sentence, I would say this:
"I finished urgent business and I returned".
Of course this doesn't really make sense, especially in context of the sentence this phrase was originally embedded in.
But I can't know if this isn't due to the fact that I overlooked something, so I reiterated on the translation.
b)
食事の途中で帰った
=> "In the midst of urgent business I returned."
Would "Being in the midst of urgent business, I returned." work as well?
とちゅうで is composed of とちゅう and で.
とちゅう is a noun and means "en route" "midway", so I guess で is in the same function as a particle as for example in:
この場所で、運動会をしましょう。
=> "Let us hold a sports day in this place."
So I would say that in phrases with "verb/noun+とちゅうで" one can always use the respective forms of "to be", even though "to be" isn't explicitely realized in the phrase (as it happens oftentimes in japanese if I observed this correctly^^).
From a functional perspective, the verb/noun attached to "とちゅうで" become attributes for "とちゅう", am I right?