You don't have to use iTunes to use an Apple tablet. We have an iPad 4th generation and don't think we've ever plugged it into the computer (where you would need iTunes). Any app purchases are through the App Store, which is separate from iTunes.
We have both an iPad and an Acer Android tablet, and recommend the Apple tablet by FAR for children. The amount of quality childrens' applications on the iPad far surpasses what is available on Google Play.
I would also, by far, recommend the fullsize iPad. Our kids are 8 and 4, and it's very easy for them to hold it and to use it. The Mini might be fine for just kids, but if you ever want to use it, you will notice the much better display on the full iPad vs. the Mini. The Mini also gives up processing power to the fullsize model (it has an older A5 chip, vs. the A5X and A6X in the 3rd and 4th generation iPad respectively). Its display is only 1024x768, at 163 pixels per inch. The fullsize 3rd or 4th generation is 2048x1536, at 264 pixels per inch. So there's a difference in physical screen size, but you also get more than 50% more pixels per inch in the fullsize compared with the Mini.
Opinions will certainly differ, but I see little value in the Mini. They're not any easier for smaller hands to use (if anything, the larger iPad is easier for them to hold), and as an adult, I can't see myself preferring the Mini to the fullsize for anything.