What is the proper term for this construction? I know exactly what the text means but I simply want to know how to look it up in grammar books.
Während an der ukrainisch-russischen Grenze bewaffnete Separatisten wüten,...
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Maybe it's easier to understand if you turn it into a main clause.
Bewaffnete Separatisten wüten an der ukrainisch-russischen Grenze.
So the participle "bewaffnet" is used as an adjective to modify "Separatisten", but "an der ukrainisch-russischen Grenze" is an adverbial for the whole clause.
Now German is flexible with regard to word order, so you can arrange the two parts in a number of ways, and it will look different in a main clause and a subclause, because the verb is in a different position:
(An der ukrainisch-russischen Grenze) wüten (bewaffnete Separatisten).
Während (bewaffnete Separatisten) (an der ukrainisch-russischen Grenze) wüten, …
Während (an der ukrainisch-russischen Grenze) (bewaffnete Separatisten) wüten, …
So the grammatical principle is just that you can re-arrange some parts of the sentence, while some parts have a fixed position. Does that answer the question? Or did you mean something else?
Your grouping is wrong. The separatists have not been armed at a particular location, the separatists are armed and they are at a particular location. This can only be gleaned from context!
Obwohl im Krankenhaus infizierte Patienten eine raschere Behandlung erhalten, sind sie durch die resistenten Stämme gefährdeter.
You cannot tell from grammar alone whether the infected patients get a quicker treatment in hospitals than somewhere else or whether patients that got infected in hospitals get a quicker treatment than patients that got infected somewhere else.