Is the liquid damage indicator activated? If you shine a flashlight in the headphone jack you should see a white/silver dot. If that dot is pink or red then it has been exposed to liquid and could explain the issue you are having. You say when charging it is easily disrupted. Are any of the pins in the USB port bent? That is a very common issue to have, especially with younger users. They often try and jam the USB cable in the wrong way and it bends the pins. I would try doing a hard reset to see if that fixes the issue. Back up your iPod first
Are you feeling any action in the button? Meaning when you press it down do you feel it going down? It could be the button itself but more likely it is the connection to the ribbon inside. Have you opened it up at all? Those can go out after a while. Check to see if you iPod is under warranty through Apple. If so you can just bring it in and they will swap it out. Make sure to back it up beforehand though.
Don't bother gluing it or taping it. The way the screen fits there is almost zero "wiggle" room for extra adhesives or tapes. You are risking having the screen sit unevenly and getting possible white pressure points on the screen and worst case scenario you could damage the LCD. Just sit it in the right place and install the screen. I have never had any issues doing it this way, just make sure the antenna is completely connected. You don't want to have to re-open it to fix.
Do you happen to have the old screen still? You can try and put that back on and see if you get the same issue. Have you tried actually charging it and unplugging it to see if that helps? Does the iPod have any liquid damage?
That is not true. I have NEVER had any iPod that would not work with a replacement screen and I have done a ton of screen replacements on these iPods. When the iPod display is not working can you see it when plugged into a computer?
Is the liquid damage indicator in the headphone jack activated? It should be white, if its been exposed to liquid then it will be red or pink. Have you tried getting different screens from different suppliers? Is the LCD connector in good shape and none of the pins are bent or missing? It sounds like it is an issue with that specific connector. Have you tried different screens from different suppliers? I have never had 3 in a row give me issues but it might be possible the supplier has a bad batch, but they most likely would have heard about it from other customers too.
Try pluggin it into a computer and see if itunes recognizes it. Did you make sure the LCD connector is in completely? Did you bend any of the pins on the LCD connector? It is worth trying the old screen to see if that one works. If all else fails try putting it into DFU mode, but fair warning it will erase your ipod.
Something is touching the LCD. It could be a ton of different things. Where specifically is the rainbow effect happening? Is it all across the top of more on one side? With every screen replacement there is usually some sort of distortion since the ipod was exposed to dust. Here is a little checklist that might be helpful Did you leave the metal rear camera shield on? Try removing that if so. You do not need that Did you line up the metal bracket on the screen correctly? Is the front facing camera sitting correctly? On the emi shield there is a little black sticky part toward the top right, make sure there is no debris there, make sure the shield is seated correctly as well. Check the back of the LCD, make sure there are no markings. If you try to put it on with a piece of glass (or some debris) it can leave a little dimple and that can give the rainbow effect.