If your Joomla user account timezone or the timezone set in Joomla's global config is set to the same timezone as the computer, it will appear that way (which may be the case if you live in the UTC timezone since that's our default setting IIRC), but no it is not based on your computer's timezone. The calendar fields in core use a filter to convert times (which are all stored in UTC) to a preferred display timezone, this will be the one set in the user's account if it's not set to use the default then falls back onto the timezone set in the global config.
I can't explain the "drift" Steve is seeing when saving things, I just know a good first troubleshooting step needs to be validate Joomla timezone settings, and it probably doesn't hurt to make sure the server's time and timezone isn't misconfigured. Beyond that you really have to get your hands dirty with the problematic sites to see what's happening.