should I replace the logic board?

I am trying to fix my friends iPod 4th Gen 40GB with mono chrome screen. The Symptions are very strange. Firstly the battery is shot (the original battery was manufactured in November 2004) and won't hold a charge. I replaced the battery with a new one from iFixit.com. The iPod now turns on and holds a charge. When i plug it in to a wall charger it charges. If i plug it in to my computer it won't be recognized by Windows nor iTunes. it says that "the device reports that it is malfunctioning." with error 45 and error 10.

Now here is the weird,

If i have the iPod plugged in, the menu freezes. if i push a button, then pull out the charger cable, the action will take place. if i am playing a song, then plug in the charger cable, the song will play for 10 seconds then stop. if i unplug the charger cable, then the song will resume playing and the Menu will respond to button presses. My question is should i replace the Logic Board? I am a techie but i don't want to invest in a $50 logic board for it not to work. I have tried different plugs and chargers.

all help is appreciated

jakthebomb