A little history first:
When my wife first got her iPod, she had problems syncing it to her computer. I got error messages about it being corrupted, then found there was a 1418 USB error. I plugged it into my computer which already had iTunes on it, and had been working fine with my iPod of the exact same type. That seemed to fix the problem, then this series of events took place:

1. I found the iPod battery had been run down very, very low, so I plugged it into the computer. It synced and started iTunes, and when it finished, I got a "please wait, very low battery" warning on the iPod display.

2. 6 to 7 hours later, (I'd left it unattended), it was still showing "please wait, very low battery", so I let it stay that way overnight. In the morning, the battery indicator showed it fully charged.

3. 10 minutes into watching a movie, it stopped with a low battery warning.

4. I plugged it back in, got "please wait, very low battery", then after awhile, maybe 10 minutes, it opened iTunes and tried to sync.
iTunes gave me an error message "The iPod cannot be synced, The required folder cannot be found", while at about the same time, the iPod display showed "no battery power remains, please connect to power". That last bit worries me. During one of the cycles where it was briefly seen by iTunes, I tried using Restore. That seemed to work, but didn't solve the problem.

5. It cycles endlessly now, tries to charge, starts iTunes, then errors, then more charging. I unchecked "automaticly start iTunes when plugged in" so hopefully it would not try to sync, and just charge itself. No luck on that as it displays "Do not disconnect" though iTunes has not started, and doesn't even try to charge.

Though it seems the battery is bad, I suspect something else may be amiss too. Is there anything else I can try before taking it to an Apple store? Since this iPod is so new, is there any chance they'd replace it with a new one?