I have seen the power issue on a few iPhone 6 and 6+'s that I've had in the shop. The power issue is almost always caused by a bad part being put on the phone. When that happens I switch out the part with a new LCD and it'll work when turned back on.
As for the wifi issue, it's possible that you disconnected the cable when removing the screen, although I don't see how you would have done that unless you took the entire board out of the phone. The Wifi cable is attached near the top right of the board. All four of it's connections are on the bottom of the board (hence why I can't see you disconnecting them on accident).
Swap the screen out with another one if you can (it might fix all your issues since it's the only thing you've said that you replaced, although it's usually not that easy), and make sure that all of your connections are secure. Other than that, I, unfortunately, have no easy answers.
Best of luck!