Try doing a hard reset. hold the home button and the power button for about 10-15 seconds (maybe a little longer than that). If this does not fix your issue then it is your lcd. But ive fixed this problem before by doing a hard reset.
It sounds like one of the chips went bad on your logic board which would cause the chips to get very hot and i mean very hot. I have burned my fingers many times repairing ipods. I suggest replacing your logic board and that should also fix your wifi issue.
In order to tell if your hard drive is bad you can go to diagnostic mode. To do that press and hold the center button and the menu button for about 5 the ipod will reboot and then press the back and center button to enter diagnostic mode. Press menu and find SMRT DAT . When you are in the SMRT DAT(SMART DATA) menu you will see Retracts, Reallocs and Pending. The number of Reallocs and Pending should be Zero. If it is not zero you have a bad hard drive also on the Retracts typically any number is ok but if it is over 250 i would just make sure that the HDD still sounds good while spinning. Also if is comes up blank in the SMRT DAT it either need a new HDD cable or the cable reconnected. If it says not supported it is most likely a bad HDD. If you have any questions feel free to ask. I work for bestbuy as an ipod technician.
You dont have to replace the whole logic board. Just buy the headphone jack itself. You will just have to solder just 4 points. However make sure that it is soldered just right and sit flushed. It's a little easier that having to replace the whole logic board and cheaper. Hope this helps you.
I suggest trying to reconnect the cable at the top left of the ipod. Also if you have opened up the ipod before make sure that the cable is not torn at all. Other than that there really isnt anything else I know other than just replacing the Bezel. If you do buy a new Bezel make sure to buy the one with the home button on it. It's a lot easier to just put on and only cost a couple extra bucks.