Apple Macbook pro a1286 utility menu window missing

Unfortunately the A1286 designation is too vague as it covers MBPro 15" models from 2009 to 2012. Try entering the system serial number here to get the exact model of the Mac:

Check Your Service and Support Coverage - Apple Support

Keep in mind many of the 15" models are known to have GPU issues and most of the GPU issues are not detectable with the Apple Diagnostics.

What is the reason you are trying to boot into Recovery Mode or Internet Recovery Mode in the first place? Chances are the cause for both failures are related.

Do you have access to another Mac? If so, what is the exact model? You can get the exact model by clicking the Apple menu and selecting "About This Mac". If you have access to another qualifying Mac, then you can try creating and using a bootable macOS USB installer instead.

I can provide instructions for creating and using a bootable Linux USB stick to see if the laptop will boot to another OS. Let me know if you are interested in this option.

If you have access to a Mac from 2008 to 2015, then you can create a bootable macOS 10.11 USB installer. You can create a bootable macOS 10.13 USB installer if you have access to another Mac from Late 2009 to 2018 assuming this 2011 laptop at some point had 10.11 installed at some point in the past since I'm not sure macOS 10.13 can be installed otherwise.

Create a bootable installer for macOS - Apple Support

If macOS 10.12.6+ had been installed at some point in the past, then remove the internal drive and attempt to boot Internet Recovery Mode using Command + Option + R. Some internal drive failures can prevent booting from external media. Make sure all external devices are disconnected as well.

"No fault found' says that under the enormously constrained conditions, in the exceptionally brief amount of time allotted, using the limited tests available, no faults were found. 

It is NOT the same as "All is Well"

On that model, be sure to perform an SMC reset, possibly multiple times, to get the Hardware back into a stable condition with the Integrated GPU as the default.