The Dell DJ has an internal battery, like the iPod does.
I have had my DJ for about a year and a half now and have not noticed any decline in the battery quality, though. I think it'll last a good long time- I charge it about once a week and use it about 2 hours per day.
Don't stress the battery life too much. Think about it like the battery for your cellphone- they last a while even though you use them a lot, and they're not terribly expensive to replace when they do die. Much like your phone, by the time the battery dies, there will probably be something better and cheaper out there to replace your player anyway. Technology moves really fast these days.
If the battery does go down, they'll replace it, if it's inside of warranty I don't think you get charged, and I want to say it's about $50 if it's out of warranty, but don't quote me on that, it's been a while since I looked that up.
It will not play anything from the iTunes music store, not directly, anyway. There's a way to get around it, but I'm not going to get into it, google will give you better instructions than I could anyway.

It will play songs in MP3 and Windows Media Player format, which you can get from most places that sell music other than iTunes. Musicmatch is the preferred music store but I think you can use Napster and Yahoo Music, also.
Finally, to confuse you just a bit more

if you are looking for kids who are kind of rough on their toys, they are less likely to break a flash drive based player like the Shuffle or anything that uses memory cards, sticks, etc. The Shuffle and other similar players don't have as many moving parts or a hard drive inside that doesn't like to be bounced around, and $99 or less on sale is a lot easier to stomach if something goes wrong or gets lost.
I'm actually thinking of buying the Dell Ditty as a backup for my DJ, I like that it has an FM radio and a screen, which the Shuffle doesn't have, and the Susan Komen Breast Cancer benefit one with the pink case and pink earbuds is really very cute.

You guys are enablers. LOL.