DVI-I to VGA working and DVI-D to VGA (and the other digital outputs) not working implies something wrong with the card's TMDS transmitter(s). Effectively, analog signals are being output properly (through the dedicated pins on the DVI-I connector, which enable a direct analog signal connection to a VGA output and aren't present on a DVI-D connector) while digital signals aren't. As a result, I imagine a DVI-I to HDMI adapter won't work either.

I don't know if these newer GPUs still used external TMDS transmitters or not...

Chances are some external voltage reference or something else along those lines is not working. Or perhaps some resistor is missing. A lot of discrete TMDS transmitters have an external resistor to set drive currents and such and I'm pretty sure the video cards have a similar mechanism present also. You should carefully look for any missing suface mounted parts under the board.

Post HQ photos of both sides of the board please.