Okay, there must be an issue in the plugin's handling of the Locations.
In "Localisation" settings you have certain Locations->Zones->Countries enabled. This will affect choices the new customers would be making when specifying their addresses, as well as the admin's choice for her specifying Location parameter in Flat Rate extension (pulled from Localisation>Locations).
Looks like the Flat Rate shipping only shows up in a store when you set the Location in the extension to "All Locations", or, specifically to the United States. If your demo customer happens to have an address from another country (Canada, for example), she won't be given the Flat Rate shipping as choice on checkout even if Places were properly set for the extension.
That said, you can choose "All Locations" in the extension to see if it helps you today.
Oh, and the flat rate cost appearing on the checkout depends on the settings you have in your specific products:
have to have Require Shipping=On, Free Shipping=Off, and Fixed Shipping Price=0 for the global Cost to be pulled from the flat rate extension.
If you specify Fixed Shipping Price >0 in the product, it will override global rate.