It's a little hard to tell from the photos, but looks like the display was ripped off. You won't be able to fix it unless you can solder and are able to solder the ribbon cable back on.
There could be few reasons for your laptop to keep turning off. It might be due to overheating or a failing battery. If it randomly turns off when unplugged, there is a chance it could be a faulty battery. The laptop also can out in 2010 it looks, so it could be overheating due to the heat sink being clogged or thermal paste starting to dry out. You could use a tool like HWinfo to check the temperatures.
Does the GPU`s fan spin? If not it could be that it's not getting power, so try and reseat the graphics card and make sure that the power cables are plugged all the way in on both the GPU and power supply side. If the fan does spin double-check that the cable to your monitor is plugged into your GPU, not the motherboard. I can`t tell if you mean it doesn't display anything or if you can get your computer to turn on and use integrated graphics. if The computer doesn't display anything and because you replaced the CPU there could be a change that your motherboard needs a Bios update to recognize the new CPU depending on how new it is compared to your motherboard.
If you go to system preferences then energy saver there should be an option that's to prevent computer from going to sleep when display is off. Maybe give that a shot, that way the computer's display will still go off, but it won't fully go to sleep.
Some easy troubleshooting steps would be make sure the PlayStation is on and look what port it's plugged into. Make sure if your TV has multiple HDMI ports that you are on the correct source that match the PlayStation.
Here is the manual for your computer. Page 127 shows what the light code mean for your computer, but it looks like you should also have beep codes. Try unplugging the computer and pressing and holding the power button for a few seconds then plugging it back in. https://dl.dell.com/manuals/all-products/esuprt_desktop/esuprt_optiplex_...-360_service%20manual_en-us.pdf
According to the guide (Galaxy Book3 Pro 16") It should have space for another SSD. If you see the FCC ID on your laptop you can also look at a tear down of the device the FCC did and look there.