Placement of object with adverbial phrase
1) is natural. 2) sound unnatural without context. In certain contexts, 2) can be used.

1) is an example of a topic-prominent sentence.

"这件事" is the topic. Everything that follows the topic describes it. i.e. "你做得越快越好". When there is a topic, you can usually translate it literally to English in the form of "As for [topic], [things following the topic]". So here, 这件事你做得越快越好。 can be translated as "As for this thing, the faster you do it, the better".

Other examples of topic-prominent sentences include


  我头疼。
  
  lit. As for me, head hurts. (This is different from 我的头疼 where 我的头 is the subject and 疼 is the verb. In 我头疼, 头 is technically the subject)
  
  笔借我一下。
  
  lit. As for the pen, lend it to me.


I will probably use 2) where I want to emphasize that you should really do it ASAP. That's why I tell you to do it fast first, then the thing that you need to do, just to be clear.


  Would "你做这件事" and "你做事情" be more natural?


你做这件事 sounds like I am telling you to do this thing, instead of some other things. 你做事情 sounds unnatural.

3) is another topic prominent sentence, it translates to:


  As for this thing, you do it.


I would say that 3) puts more emphasis on "you" while 4) puts more emphasis on "this thing".