I'm wondering why on earth did they use a gold wire to resolder the power button cable. Replacing the part is the proper fix for that.
For the home button cable broken, the solution to get the home button working again is to either get the home button (comes with cable) replaced or the wire resoldered which is literally near impossible to do if you want to keep the touchID functionality.
Because the fingerprint scanner on the original home button is linked to the logic board, a replacement home button will not have the finger print scanner working.
The repair shop should fix the phone at least as to the condition the phone was in when you bring it in. Sounds like the repair shop broke a few cables which shows they may not have adequate experience in doing iPhone repairs.
A teared / broken home button cable will not stop the iPhone from turning on if you keep it unplugged.
I don't think you can check if the display works with a oscilloscope / voltmeter. The only way I'm pretty sure is to try another screen.
Can you shine a flashlight very close to the screen to see if it is displaying anything (if you just tried to turn the phone on you may need to wait a bit for it to boot up as it will show apple logo first usually).
Have you tried connecting it up to the computer and see if the application iTunes detects the device?