mbp needs nvram/pram reset to boot

First off unless the anti-virus is required by a school or employer you should uninstall Sophos by following the developer's instructions. AV and other cleaning apps are not needed on a Mac and usually cause more problems than they solve plus they impact system performance.

What is the condition of the battery? You can check it by Option-clicking on the battery icon on the menubar. Anything other than "Normal" or "Good" indicates the battery should be replaced.

Is the laptop using the power adapter?

Does the battery have a charge on it?

Is the laptop being shutdown or put to sleep?

Is this a 15" & 17" model? These models are known to have GPU issues which can cause the gray screen issue when booting. I've also seen some of them have trouble powering on as well. If it is a 15" or 17" model, then it is possible to force the laptop to use the Intel GPU instead.

Try an SMC reset.

Run EtreCheck and post the report here using the "Additional Text" icon which looks like a piece of paper.

I would start by running the built-in Apple Hardware Test while plugged into A/C power. If any mysterious codes pop up that would tend to confirm something is going on. Definitely tick the box to run the full memory test as well. Any positive results means it's time to replace that 2011 MacBook Pro.

https://support.apple.com/en-us/HT201257

You can also pull out the RAM and just try various single-sticks, or if you have the original memory still you can swap. I don't think it's a RAM issue and the AHT is pretty good about detecting failing memory, but it's still worth a try.

There is a possibility the issue is the fan is really, really, gunked up and can be blown out with compressed air. If the system isn't cooling properly that would cause slow downs, over-heating, and the various other problems. It's also possible that the fan itself or the thermal compound for the fan has simply broken down causing the system to overheat. You can use the Mac Fan Control App to run the fan at full maximum speed by force to see if this reduces symptoms.

If all else fails you can set up an NVRAM reset through terminal to pop-up every time the system is started. This is a total band-aid measure, but if you had the following command set to run in a startup script that might make it a bit easier to cope with for a while:

sudo nvram boot-args=”-p -r”

It does sound like though that (unless it's purely a RAM issue or a fan issue) you're dad's dealing with a failing logic board most likely.

And if it is the logic board, if the RAM used exceeded the required specs for the 2011 MacBook Pro (PC3-10600 DDR3 1333 MHz) and satisfies the specs for a 2012 MacBook Pro (PC3-12800 DDR3 1600 MHz) then upgrading to that newer used Mac would at least let you carry over that 16gb of RAM along with the SSD to something newer but still very inexpensive since he's reluctant to upgrade at all.

PaperSpecials wrote:

When it happens the nvram/pram reset is usually what fixes the issue and lets me boot, if not I already do the smc reset.

This tends to point to a Logic Board issue.

It is a 15" version, I did read about the switching to the nvidea gpu but it involves soldering an resistor?

Added: I just noticed at "about this Mac" it states it used the video gpu 512mb, so it is not on the amd.

To force the laptop to use the Intel GPU requires the use of gfxCardStatus. However, if your laptop is unable to even boot, then you may need to force the Intel GPU by modifying the NVRAM settings which I believe requires using a Linux boot disk. I'm not familiar at all with the latter option and the instructions may differ for different models. There are a few posts on these forums with detailed instructions. As for replacing a component on the Logic Board, that fix would be for trying to permanently fix the issue.

https://github.com/steveschow/gfxCardStatus

NVRAM option:

https://apple.stackexchange.com/a/297114

<etrecheck report.log>

If you do get the laptop booting again, the uninstall Sophos anti-virus by following the developer's instructions. AV software and cleaning apps are not needed on a Mac unless they are required by your school or employer. AV & cleaning apps usually cause more problems than they solve and they impact system performance.