Torben,
A couple of remarks that might help:
1) With 'test_request = true' all your authorization requests to Authorize.net will be approved. But only authorization requests, all other requests such as 'CAPTURE' will fail.
2) If you have the Authorize.net Gateway mounted at /authorize-gateway then you can check the log at /authorize-gateway/admin. No need to run queries in psql.
3) The customer who ordered won't get a failure notice unless the transaction for the order has been tried and encountered a) a permanent error or b) ran into repeated temporary errors exceeding the PaymentRetryPeriod (an ecommerce package parameter).
4) Upon placing an order the customer receives a confirmation email as soon as the order has been accepted. An order is accepted when the transaction is approved or when an transitory error occured while requesting approval for the transaction.
5) The next email a customer receives is when the order has been schipped. Try shipping these orders and see if a) the authorize-gateway is being contacted (the request should fail, see 1).
6) It is odd that no e-mails went out. At least the confirmation e-mail should have been mailed.
7) To test I would turn the test_request parameter off. Yes, you are now running live transactions. But as transactions are not processed by Authorize.net until the end of the day (configurable in Authorize.net) you can cancel all capture requests that have been approved before they get processed by Authorize.Net. This will give you far more realistic test results.
8) Authorize.Net changed the URLs of their documentation which broke most of the links from authorize-gateway. All links were publically accessible at the time of writing. Try https://secure.authorize.net/docs/ instead.
It seems to me that Authorize.Net has never been contacted or there would have been entries in the log. What exactly when wrong I can not tell from your description. Place trace message in the code and see where it fails.
/Bart