Learn Swedish: Vi debiterar arvodet i den mån vi slutför uppgiften. - We charge the fee to the extent that we complete the task.

The phrase i den mån (to the extent that / insofar as) is a subordinating conjunction. It is very often written as i den mån som, but just like the word "that" in English, the som is frequently dropped to make the sentence flow faster, which is exactly what happens here.

Is arvodet the standard word for any kind of "fee"?

Not quite. Ett arvode specifically refers to a fee, honorarium, or remuneration paid for professional services, like those of a consultant, lawyer, or freelancer. If you were talking about a general administrative charge or a membership fee, you would use en avgift instead.

Why use slutför instead of a more common word like slutar or avslutar for finishing the task?

Slutar simply means "to stop", which doesn't imply the task was actually done. Avslutar means to finish or conclude something. Slutför (literally "leads to an end") strongly emphasizes completing or fulfilling a goal successfully, which makes perfect sense when you are charging a professional fee for it.

How would the word order change if we added inte (not) to the second half of the sentence?