With that one if I remember correctly you have to toggle the lock button back and forth while holding the volume down button and press the power button. If not volume down try the up button instead. You should get the android symbol with a red triangle. Let me know how it goes and toggle quickly.
Awesome you got picture. The next thing I'd try you are going to need the controller for and a USB drive. You're gonna wanna go here https://www.playstation.com/en-us/support/system-updates/ps4/ and all the way at the bottom you'll find info for reinstalling your software not just updating. You'll want to click the download now text follow it step by step. Then after you're done plug in your usb drive and put the ps4 into safe mode to start reinstalling or reinitializing software. Let me know if this helps.
Yes you can just swap out the screen it usually comes with tools, but I like to use a spudger to open the phone. I advise using a little heat near the home button connection. The flex cable is pretty flimsy and sometimes the adhesive can have a snug hold. You do not want to damage the home button on disconnection. I strongly advise using one of the step by step guides on disassembly and putting your screws in order. Let me know if you have any other questions man.
If you have a hot air gun you can usually turn it to 300 degrees cels with half air flow apply heat lightly in circles for about 2 minutes then use a spudger to slide between the rim and glass.....replace with an ultra thin guitar pick and slide around it until it loosens completely. Let me know if any issues.
Have you cracked it open yet? Seeing is believing. It's hard to tell what's bad without doing so. If you want you can check your connections on the outside for corrosion first. Update (10/18/2017): Alrighty.....that's some pretty bad corrosion and you have some components that are fried. I'm honestly not sure that will be a fixable project to be honest because there are so many dead parts. As soon as you start cleaning it they may fall off.