In all usages of da words I came so far, it goes along the lines of the idea first and then the da words is used to re-reference that.
This is not outright wrong, but covers only half of the truth. You can use what you call da-words in a way you use a Demonstrativpronomen: as a sort-of placeholder for some longer construct. It doesn't matter if this longer construct it refers to comes before or after. You could rephrase above sentence like that (leaving out the writing, which is irrelevant for this):
Ich bitte Sie darum, ab dem nächsten Monat zu Hause arbeiten zu können.
Ab dem nächsten Monat zu Hause arbeiten zu können, darum bitte ich Sie.
Bitte gestatten Sie mir die Arbeit von zu Hause ab dem nächsten Monat.
In the first variant you establich the topic of your request with "darum", then specify what "darum" stands for in the Relativsatz. In the second version you first establish this topic, then sum it up with "darum".
In the last version you don't need the placeholder because you directly specify the topic of your request.
Here is another example for the same construction with "damit". It sums up what is said afterwards in the Relativsatz:
Dem Armen wurde damit geholfen, ihn reich zu machen.
The reason why constructions like in the first and second variant are used is because they make the statement of the core point easier: I ask you for - with what "that" represents to be explained separately. If this explanation is before or after is of no concern. Also, the pauper was helped with - and "that" being detailed later.
You have similar means in English:
He did it. "It" being a complicated task needing a long-winded explanation of what it constituted.
or
It was a complicated task needing a long-winded explanation of what it constituted. He did it.
In both cases you sum up the long version (before or afterwards) by the "it" in the main sentence.
