The "/var/www/html" should not be there (so this would explain the bad relative path that would give me a permission problem), but how did it get there if my uploads_path is "wp-content/uploads"??
The uploads_path controls where the file goes, but the other uploads option controls what the URL looks like. See the Settings->Misc screen.
When I try to change the "root" upload path, it does not change anything in the path for the images associated with individual posts - these seem to be "preserved" with the individual post.
No, they're preserved for the individual attachment. Changing the upload path doesn't magically make your files actually move around or something. Changing the upload path causes new files to go to that new location. Old files stay where they were.
What I am worried about is that the WordPress code may have a "/" in front of some path built from the database instead of using "../" for making it relative.
Relative links won't work, because the URL is not a fixed entity, it changes on single pages, on category archives, on the main page, etc. That's what the Permalinks do, after all.
WordPress uses absolute links at all times, in all places.
It is only when I try to "insert into post" that I get the blank screen.