Can 下次 mean "again"? I found a sentence on tatoeba that isn't quite the translation I would expect.


  I will not do it again.


Is translated as...


  下次我不做了。


I checked mdbg.net to be sure, and 下次 is only translated as "next time", not "again". Is there another shade of meaning that I'm missing, or would this be significantly better?

我不再做。

Does 了 have any effect on my question, or is it just signifying a change in state?
The strict answer is no.

下次 is a shorten form of 下一次 which means next time.
下 carries the meaning of subsequent, the following or the next.
一次 carries the meaning of one time or once.

So, taken together, it means the next time.

But, for the next time to occur, the same or similar situation has to repeat itself (again). Therefore, you are not wrong to translate in the manner you have quoted above.


  I will not do it again.


is equivalent to


  I will not do it the next time.


Or it can be written redundantly as:


  I will not do it again the next time.


In Chinese, you can also write the three sentences above to mean the same thing:


  我不再做就是了。
  
  下次我不做了。
  
  下次我不再做就是了。


Although the words don't necessary have the same meaning, they can be written to mean the same thing.

The second part of your question is quite difficult to explain. My opinion is that 了 is used to indicate the closure or completion of an expression. I am not a language expert but I believe that the term perfective aspect may be useful.