Late 2011 Macbook Pro not able to go into recovery and fsck not working

Even though the Logic Board & GPU were previously replaced you could still have a bad GPU on this board.

You can try forcing the laptop to use the Intel GPU by using gfxCardStatus.

https://github.com/steveschow/gfxCardStatus/releases/tag/v2.4.4i

If you cannot boot normally to install gfxCardStatus, then check out these links for alternative methods of forcing the laptop to use the Intel GPU:

http://dosdude1.com/gpudisable

See the latest news section of this link:

https://github.com/steveschow/gfxCardStatus

There is also a chance you could have a failing hard drive. If you cannot boot into macOS to use DriveDx, then using another computer you can create a bootable Knoppix Linux USB drive which can be used to check the health of the hard drive. Use the downloaded Knoppix .iso file as a source for Etcher (Mac/Windows/Linux). Option Boot the Knoppix USB drive and select the orange icon labeled "EFI". While Knoppix is booting the computer may appear frozen on the boot picker menu so give Knoppix lots of time to finish booting.

If Knoppix boots to a desktop, then click on the "Start" menu icon on the lower left corner of the Taskbar and navigate the menus to "System Tools ---> GSmartControl". Within the GSmartControl app double-click on the laptop's hard drive to access the drive's health information. Post the complete report here.

If Knoppix only boots to a black screen with white text and a command prompt, then it may mean the GPU is faulty. If you use the following command it will access the health information on the laptop's hard drive (assuming only one drive is installed or connected to the laptop):

sudo  smartctl  -a  /dev/disk0  |  less

Use the up & down arrow keys or the Trackpad/mouse wheel to scroll up & down the health report. Take pictures of the report and post them here. Pressing "q" will exit the health report and return you to the command prompt where you can issue the shutdown command or just hold the power button to turn off the laptop:

sudo  shutdown  -h  now

You can give this a try which should work on your model:

Network Recovery of Lion Through Yosemite on a Clean Disk

1. Restart the computer. Immediately, at or before the chime, press and hold down the Command-Option-Shift-R keys until a globe appears.
2. Wait, patiently, 5-15 minutes until the Utility Menu appears.
3. Select Disk Utility from the Utility Menu and click on the Continue button.
4. After Disk Utility loads select the drive (usually, this is the out-dented entry with the mfgr.'s ID and size) from the side list. Click on the Erase button in Disk Utility’s toolbar. A drop-down panel will appear.
5. Set the partition scheme to GUID then click. Set the Format type to Mac OS Extended (Journaled.) Click on the OK button and wait for the process to finish. Quit Disk Utility and return to the Utility Menu.
6. Select Install OS X from the Utility Menu and click on the Continue button.

What is the exact model of your laptop? Is it a 15" or 17" model? Maybe you are encountering a failing GPU which is common with the 15" & 17" models. There is also a good chance your hard drive may be failing. What happened when you tried Verbose Mode? Did you notice if there were any "I/O errors" listed in the boot log as it scrolled by which would indicate a failing drive?

What version of macOS are you running on the laptop?

How did you boot into Single User Mode? Standard Single User Mode will boot with a black screen and white text and display instructions on how to run the file system check and how to mount the root of the boot drive as read+write. Standard Single User Mode defaults to mounting the root file system as read-only.