Direct HDMI output at the start up?

USB to HDMI converters require special driver. I never heard of an Android device supporting such a converter. The typical way to connect a Android device to a TV is using Wifi and screen cast technology.

There are some device that support MHL (re-use the USB-Port for a HDMI connection) but the number of device that support this is not very high and depends on the used chip in the phone. Anyway the USB port would be blocked in such a case and can't be used for USB devices.

Edit: Based on the additional info your provided (unlock possible, touch still works) the following may work:

1. Unlock the phone and connect it via USB to a PC. You should get access to your sd-card folder.


2. Use the screenshot feature to get the current screen content.


3. Open the screenshot on the connected PC


4. Try to tap on the position of the function you want to execute


5. Goto step 2

Using this you should be able to "use" your device in a very very basic way. Therefore what ever you want to to with it (e.g. exporting data) might be possible this way.