Automatic???

Since WordPress 3.7 minor point releases have been automatically updated. It actually was a major feature for 3.7.

See the release announcement for details.

WordPress 3.7 “Basie”

Only minor release versions such as 3.8 -> 3.8.1 will be automatic. A major release number change will not be automatic so 3.8.x will not upgrade to 3.9.

a) What if there are changes that badly affect my site? I like to do these upgrades on my test site first. I don’t want to be upgraded automatically.

Fair enough and you do have a choice. 😉

If you want to disable automatic core updates then add this one line to your wp-config.php file.

define( 'WP_AUTO_UPDATE_CORE', false );

And that will disable that feature on your installation.

b) Although the site’s Dashboard says it is now 3.8.1, I can’t find any files whose dates show that they have been altered. Perhaps only the Dashboard announcement has been changed and not the actual functionality of the site?

Updates to core previously downloaded the entire WordPress file set but now only the altered files are downloaded. That makes for a smaller download and safer upgrade.

You can see the list of files via the article.

http://codex.wordpress.org/Version_3.8.1#List_of_Files_Revised

c) Even odder is that my private test site (on my laptop only) also says it has been updated to 3.8.1!

Yep. All WordPress 3.8 installations will eventually automatically update unless the option is disabled.

You an read about all of the options in great length at this link.

The definitive guide to disabling auto updates in WordPress 3.7