1- Battery depleting very fast could be short on the board, or bad battery. Replace with new or at least known good to test. I highly recommend iFixit batteries because otherwise bad batteries and repackaged/used/worn batteries are the norm, not the exception.
2- Battery turning off at 40% for example and turning back on at that same percentage or something significantly higher than 1-4% is:
2-a- Either a firmware bug (known iOS 10 issue, update or even better restore from DFU to the latest iOS version after backing up. 10.2.1 and newer are supposed to fix it),
2-b- Or a bad battery of 2-a doesn't solve it. Go back to point 1.
3- Your charging port is bad and needs replacing.
4- Your charging circuit is bad and needs replacing. Board level issue.
[1-2-3: when you connect the charger, you have to be able to see a battery icon or battery+cable icon. 4-5: you could be seeing those, or not].
5- You "think" your iPhone is not turning ON because nothing appears on the screen; but when you connect the iPhone to a computer, iTunes detects the iPhone and tells you to allow access/trust. Or if already trusted in the past, you see the iPhone details in iTunes. In which case you damaged the display circuit when replacing the battery. Or you damaged the screen assembly.
5-a- Try a known good screen. It has to show an image, so the test screen can be an old cracked one. As long as it is known to show an image.
5-b- If you rule (5-a) then either you backlight is damaged, or your display circuit is damaged (connector, filters, chestnut to name a few), or you caused long screw damage to the bottom left screw hole of the shield holding the screen cables in place.
[4 and 5 all require micro soldering tools and skill to trouble shoot properly and repair].
I hope I didn't forget anything but the above are the most common issues we see coinciding with a screen or battery replacement.