Does our DNA change during our lives?
To quickly answer your question, yes DNA changes over the life time of many organisms including humans.  You have a whole host of mechanisms in your body that try to prevent your DNA from changing, but they are not perfect.

A good example of this is DNA degradation due to aging.  When discussing aging in biology, we use the technical term senescence.  Of interest here is the shortening of telomeres due to mistakes made in your cells as they continue to divide (copying the DNA to each new daughter cell).

In a grown adult, there's not some master well of DNA somewhere such that if you change it there it will immediately change the DNA throughout your body.  Most of the cells in your body have DNA in them (red blood cells are just one example of cells that don't have DNA in the traditional sense). Most of these cells can and will divide over time (mitosis) to produce new cells with copies of that DNA.  However, there are also plenty of cells that don't divide often or at all in healthy adults.  If one is concerned about schizophrenia, it would be important to note that many neurons in the CNS are in this group of static cells.

Considering environmental effects, there are plenty of things that can change your DNA. Radiation, gamma and x-rays in particular, do a great job of disrupting, degrading, and occasionally mutating DNA.  The class of chemicals and physical agents that can change DNA are called mutagens.  This shares a root with the verb mutate, which is when something is acting on genetic material to cause it change.

You will note that I said genetic material, and not DNA.  Without getting too far into the debate of what is "alive," DNA is most certainly not the only "construction protocol" for earthly organisms.  RNA is one common alternative, but there are weirder ones that are hard to describe like prions.