"When an E. coli goes through 10 cycles of division, all that results is you have 1024 identical copies of the same E. coli. In what sense is information processed, rather than simply duplicated?" 
Nice question, by the way (easy answer, though)!
When the E. coli (or any other organism) goes through cell division, its DNA system processes the mathematical instructions contained in its genes.
Because these instructions are processed instead of merely expressed (i.e. copied/templated/et al), all genes therein will be replicated in their entire-ity EVEN THOUGH some of those genes may be turned off.
But a blind "expressive" system will merely copy something (think of a photocopier). To turn off a part in an expressive system, that part of the template must be entirely deleted, overwritten, or erased. Thus, whatever is copied in a blind, expressive system must be active / turned ON.
In contrast, an active processing system can turn off part of itself without deleting it, without erasing it, without overwritting it, etc. A system that processes instructions can be told to copy, but not execute, the commands/data in question.
Thus, your E. Coli example will copy ALL of its genes even though some of those genes are turned off.
Because these instructions are processed instead of merely expressed (i.e. copied/templated/et al), all genes therein will be replicated in their entire-ity EVEN THOUGH some of those genes may be turned off. You're trying to have it both ways here. If you're talking about replicating DNA, then the thing that does the replication is the cellular machinery, not the DNA. On the other hand, if you're talking about replicating the cell, then the turning off/on of genes is part of the replication process, and can't be separated from it. To replicate a cell, you need to do a lot more than copy the DNA code.
A cell is self replicating, and control of gene expression is part of that process. A piece of DNA is not self-replicating.
And I'm missing totally what this has to do with evolution.
its DNA system processes the mathematical instructions contained in its genes.You are seriously confused between things and models of them. I guess you think regular computers process mathematical instructions? No, they go through a sequence of electrical states just like any other electrical device.
Math enters the picture only when we abstract the process. IOW there is nothing inherently mathematical about either computer processors or gene expression but they can be modelled mathematically.