I understand that cell DNA changes over time due to damage from a variety of sources, and that there are mistakes made during copying. What I'd like to know is, say, for the average person living in the U.S., exactly how common this damage is.

Is it possible that such errors are so common that the idea of a person's "genetic code" is really more just an ideal average of all the individual cells' DNA taken together, rather than something that many cells actually have exactly? In other words, what fraction of our cells have DNA differing from what we began with (whether the difference is apparently critical or not), and to what degree?

you are correct - the rate varies between tissues/cells and individuals and will be composite of environmental and endogenous factors (the most important in normal individuals as you say are error rates in mitosis). see below

now we are starting to mass sequence the genomes of large numbers of individuals it shows the differences starkly (so called polymorphisms rather than mutations, since the vast majority have no known functional effects). Thus there is no absolute "to start with" - we are all individuals with differences.

http://www.nature.com/news/2009/090827/ … 9.864.html