It would be great if there were an app that could simplify the process of recording your own versions. (Is there such an app?) It would be interesting to try this with songs and other sounds instead of just a speaking voice as well.
If you want to emulate this purely in software there are convolution reverbs[1] available for both Pure Data[2] and Csound[3] that you could probably re-purpose / script to do tens or hundreds of iterations on any input material you wanted.
Note: the linked examples are both based on "partitioned convolution" algorithms[4], which allow lower latency than standard convolution algorithms, and would probably be a better choice for running many iterations in a reasonable amount of time.
I may play around with this when I next have some spare time.
Also, if you are using OS X, there's a spectral processing program called SoundHack that let's you convolve (along with several other processes) any two sound files. I am not sure where to grab the latest version, I believe development has been abandoned but it was a stand-alone freeware that one of my '" Intro to DSP" courses used to introduce certain functions.
Nope. DNA copying is surprisingly reliable, and there are several mechanisms that prevent this kind of degradation.
Also, as soon as a serious bad (deleterious) mutation happens, the resulting individual won't be able to reproduce further -- this selection process ultimately prevents this from happening.