iPod 4g wont sync music

My iPod 4G hasn't been working correctly. The battery is most likely fried because it doesnt last for long and it charges quickly. It only lasts for an hour or so. It also has a problem with putting new music onto it. I tried backing up my iPod and then syncing the music, but it wont do it. I put the music onto my new laptop through a CD and it went onto my iTunes Library, but when I tried putting it onto my iPod it went through the whole process of syncing, but nothing happened. Nothing popped up, no errors, or anything. My music also skips around. I went to play a song and it completely skipped over it. I tried again and it still didn't work. Now that song works(I dont know how I fixed it), but now it's doign it to other songs. I've had this iPod for about a year now. It's getting frustrating. Can you please help me solve these problems? Thanks.