There is no reason to buy 144hz with a gtx 1060 imo ... Idk what games you want to play but the gtx 1060 wont be able to have stable 144 fps in new games.
If you press the power button, nothing happens? If that's the case I think you accidently plugged the Power Switch plug from the motherboard while cleaning your pc. Please, check if it is plugged in :)
What is your cpu running at in percentages? (load) If the cpu is running 100% and your gpu running at 60% then your CPU isn't powerful enough for the gtx 1060 3gb.
Yeah your gtx 1070 won't get 144 fps in most games with high graphics. You could try to lower your settings in the graphics but still then it'll be hard to have stable 144 fps.
Not really, you could clean your pc a lil bit before you replace those components. The only thing you have to do is to download drivers for your new gpu. :)
This could also be a short circuit somewhere in a component. Or the PSU is indeed faulty, unfortunately it's hard to test so you should try to replace the psu if it the problems still occurs you should try replace the mobo, etc, etc.