This would allow us to predict the future and potentially allow for time travel as well. Seems unlikely from the implications alone, but let's see what peer review of the details will tell us.
It doesn't actually, they discuss this in the article. The laws of thermodynamics impose an arrow of time via boundary conditions, and something similar would apply here.
Also, the conspiracy may be a good thing. If you have self-replicating and mutating code on your computer you should put some sandbox around it, or it might evolve to buy more RAM on ebay. Or, more realistically - will crash the system.
Well that's just inevitable. You're never going to see a naked singularity, for instance.
Retrocausality would probably work in the same way.