There was a brief explanation that I don't think I quite caught.... it's too easy to track bodies in the future? Or something? If murdering people in the future is so hard, why did they shoot the chinese wife? If they can get away with shooting people and just need a place to dispose the body, why don't they send the bodies back in time pre-killed? Would seem to be a great way to avoid loose loops...
That's where I began having problems, too. Though after thinking it over for a while, my guess is that the mob figured that with Joe's wife dead and Joe missing, the authorities would assume Joe killed her and not investigate things further. Still a bit of a stretch though.
And the Loopers just throw the bodies in an incinerator, why can't the mob do that? Okay, maybe it might raise some red flags when the body's tag stops transmitting, but that happens anyway when they get sent back in time.
And if the Looper has to be killed in the future to prevent them from squealing about the time travel, why are the people sent back as recentally as thirty years? Going back say sixty years and you have less of a chance of your Loopers still being alive in the present day. Especially considering the kind of lifestyles they seem to live. I suppose it's possible the time machine could have limitations and they can only go back thirty years, but it wouldn't have hurt to throw in one line saying so.