OK, I fixed it. It was the fuses.
After changing them as shown in the attached image, the 2 displays I had connected showed some beautiful digits.
I originally just flashed the hex file called HP3457Oled_HP3478_ICONS.hex - and then tried adjusting some fuses suitable for 8MHz and 3.3V, but didn't have any luck.
Opening the project in Atmel Studio (I have 7.0 installed), I noticed it started in the Debug configuration. Compiling failed with an error that F_CPU wasn't defined.
I then changed to Release configuration where F_CPU was defined to 8000000 (under Project Properties, Toolchain, C Compiler, Symbols).
Building the Release config worked without error, and I then flashed the ATmega8L on my board using the generated elf file.
I then didn't seen any random dots on the display on power up, and after plugging into the 3478A display cable it worked.
Miti, maybe this will work for you too. Let me know if you need any tips, but I'll be offline for at least 8 hours.
Edit: Updated image with new fuse values of DE A4 per Miti's comments.