"Please finish writing the report as soon as possible."
and more generally can someone break down the word order and what's optional or open to different ordering?
There are different ways of expressing this but in this sentence every word is needed in its place.
请 (please) + 尽快 (as soon as possible, as fast as you can) + 把 (supporting word)+ 报告 (report) + 写 (write) + 完 (finish)。
把 has to be in that place and is a 介词 which helps form phrases.（介词总是同其他的词组合在一起，构成介词短语，做定语、状语和补语。）
"Please finish writing the report as soon as possible." On another note, I think "Please complete the report soonest possible" should also be accepted.
"Please complete the report soonest possible" isn't correct in English. Maybe you mean "at the soonest time possible?"
I do; it is colloquial, not proper, strictly correct English, sorry about that.
I answered the same and was marked wrong, but as a native speaker of Chinese, it sounds totally acceptable.
You totally can. Your average Mainland Chinese is going to use the less-formal construction you have used.
@bobking12 I think its grammatically incorrect to have a modifier between the verb and the object in Chinese. In your answer the modifier 完 is between the verb 写 and the object 报告. For the sentence to be grammatically correct you have to use the 把 structure as they did in the answer.