Justin, thanks for the reply.
I initially set up the server with apache as the owner of the /html folder. While troubleshooting problems I changed the owner to root, but it did not fix the problem. I have now changed it back to apache (I do not plan to run another site from this server). So I am assuming that this should be secure enough to run this WP site?
I have confirmed that I am not running Fast_CGI. Also, I confirmed that PHP is being run as the “apache” user.
With all that said, I think I am good as far as the site working and functioning under the correct ownership. I am able to update plugins, but I still get the “inconsistent file permissions” message mentioned in my (1) problem. I plan to let someone else manage WP once the server is up and running and I need for this feature to work. That message is what led me to change permissions to begin with. Any idea?