I have only one payment method setup - Authorize.net - and I checked the option to preselect payment methods when only one exists. There are no countries or shopper groups assigned to the Authorize.net payment method yet I still get No Payment Method selected at checkout. WHY???
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How long ago did you setup Authorize.net? You have to have them set your account to live mode, they have to look at your site and tell you recommendations before they will just let you start paying, also they don't handle your money, they find a provider for you-perhaps this isn't entirely setup with Authorize.net?
Also, if you don't have an SSL it won't work unless you purchase the SIM module from the Virtuemart Store.