I'm posting an answer to my own question (but I won't mark it as the official answer because that would be rude) to sum up what others have said and also what I heard from Canon when I called technical support.
As others have mentioned, the answer is, "No." The presets are not modifiable and the camera will not remember the last used sub-settings (such as flash-off) once it gets turned off and then on again.
Using P and Av are the best work around for this. In my case, I'm using P with ISO:Auto as my default shooting mode and then switching to Av if I want control over depth of field.
I have also started shooting in RAW + JPEG (RAW + L) instead of just JPEG (L) so that I can fix white balance and other things that a preset might adjust for me in post.
I hope Canon will offer a setting in the customization menu (in entry level, lighter body models) that lets the owner of the camera prefer available light over using light from the flash. I let them know my thoughts. Maybe maybe they'll take my suggestion and put the preference setting next to the no-beeping setting.