I have opened a second ticket on the Delock support even if the problem seems to come from the GPU...
Finally, I have correctly installed AMD driver on Linux Mint and the image appears like Windows on the login screen but the maximum resolution showed is 1024x768 worst than Windows...
1: can't see the POST
2: can't choose which OS to boot
3: can't get 1920x1200
The Delock answer:
The problem will probably caused by an incompatibility between your graphic card and our adapter and can not be solved from us.
The adapter is a plug-and-play device and does not require any additional driver software, because the default drivers on Windows 7 and the kernel drivers on Linux (off kernel 2.6.38) works in our tests.
The cause is most likely caused by an incompatibility, can be derived from the fact, that the boot / post is not correctly works.