You'r ipod needs a new battery. you can buy them on ebay for relatively cheap, with shipping it should cost around $10, installing it isn't to hard however you will probably need a soldiering iron. I would provide the link to an fixit guide and an ebay auction for the battery but you haven't told us what generation the ipod is.
I don't believe that you can fix the problem, likely a piece of dirt got into the volume control and is blocking the contacts however I don't believe the volume controls can be opened without destroying them all together. It may still be possible to get them replaced by Apple, if the ipod you are using them with is still under warrantee (all new ipods come with a 1 year warrantee which will cover you'r ipod and the in-ear phones you'r using it with, however can be extended to two years by purchasing the applecare program) to check if the ipod is still under warrantee input you'r ipods serial number into this webpage https://selfsolve.apple.com/GetWarranty.... Hope this solves the problem!
Yea this sounds like it's a cheap knock-off. Turn you'r "iPod" over and read the serial number and put it into the apple site to see if the serial number (if there even is one) is valid in the apple database. Input the serial number into the applecare warranty page on the Apple site below, if the ipod is a knock-off then the serial number wouldn't be valid in the database. https://selfsolve.apple.com/GetWarranty....
Hey I have the same problem with my 4th gen nano, I pulled it apart yesterday to replace the screen and it looks like the cable has been corroded on the top 2 pins. We also considered that the logic board was failing to provide power to the screen, but seeing as you have the exact same symptoms I doubt its the power supply. Unless you can replace that cable I don't think the ipod can be saved. I defiantly think that the cable is the problem here, does anyone more knowledgeable know if the cable can be replaced? if so where could I find the part?