Learn Russian: Я быстро сделал работу.

This is the difference between Perfective and Imperfective verbs! сделал is Perfective, meaning the action was completely finished with a clear result: the work got done. If you used the Imperfective делал, it would mean 'I was doing the work' (focusing on the process, without saying if you actually finished it).

How can you tell from this sentence if the speaker is male or female?

You can tell by looking at the past tense verb! сделал ends in a single -л, which is the masculine ending. If a woman were speaking, she would add an 'а' to the verb and say сделала.

Why does the word for work end in 'у' (работу)?

The dictionary form is работа. Because the work is the direct object of the sentence (the thing receiving the action of being done), it needs to be in the Accusative case. For feminine nouns ending in -а, that ending changes to -у.

Can I change the word order and put быстро at the end of the sentence?

You can, because Russian word order is flexible, but it is most natural to put adverbs like быстро (quickly) right before the verb they describe. Putting it at the very end (Я сделал работу быстро.) might place a heavy, special emphasis on exactly how fast you did it.