MAJOR UPDATE 3: IT'S WORKING!
EFI was outdated and caused Linux not to DHCP for some reason! I solved this issue by installing High Sierra on to SSD using USB adapter using my other iMac5K. After plugging it back into SATA in 2011 iMac, I enabled shared desktop in Sharing Settings in macOS and put SSD into 2011 iMac. It surely booted and I confirmed old EFI firmware. Updated EFI by installing security updates from AppStore. You can confirm your EFI firmware using tool SILENTKNIGHT (google it). Booted into Linux USB after 5 consecutive PRAM resets (5 chimes in a row), DHCP and SSH worked almost instantly (I think i had to press ENTER blindly 3 times).
NOTE: I think on older EFI, the fans were spinning like crazy during Linux session, after update, they do not. In macOS I still prefer to use Macs Fan Control app.
ORIGINAL MESSAGE: I'm having hardest time. For many hours been trying to figure it out. Read this thread's wiki multiple times....
iMac (27-inch, Mid 2011) - 3.4 GHz Intel Core i7.
Modding to have GTX 880M. Pulled everything out, modded the GPU cooler and X-bracket with Dremel, applied k5 pro thermal paste, installed 880, refreshed CPU paste, put everything back together.. And black screen!!! I can boot into High Sierra and remotely connect to it (screen sharing), control macOS at 1280x1024. I even ran benchmark Unigine Valley, it runs!
I cannot for the love of anything, get it to DHCP in Linux USB stick. Tried 4 USB sticks, SD card. Tried the blind method of turning on DHCP (hit N, then hit ENTER 5 times, etc), tried following along on my other iMac5K. It just doesn't get the IP address. Tried 3 different routers, factory reset them. DHCP has been randomly issued IP address a few times, but it wouldn't let me SSH into it (or even ping) - saying host is not available or timeout. Tried USB-ETH, tried TB-ETH. Nothing. iMac5K works (has screen that I can see) it shows up in the router devices list. The 2011 DOES NOT! Remember, it runs macOS fine, gets IP, there is GPU acceleration, but NO IMAGE on screen.
Help please... I don't know if the GPU is somehow faulty or if something is wrong with Ethernet...
Note: 2 out of 4 LEDs on logic board are on when iMac is powered on.
UPDATE 1: I had outdated EFI firmware, doing software update over screen sharing updated it to the latest 87.0.0.0.0 will test LINUX next.
UPDATE 2: Booted into Linux USB after 5 PRAM resets (5 chimes in a row), now it shows up on router and has PINGable IP. Trying to FLASH NOW