No App Store on my child's iPod

I bought my 7yr old daughter an iPod for Christmas (no judgement trolls please) and I got it all set up with her own apple ID and music and apps etc, charged it and then put it back in the box so that it was ready to go for her when she opened it Christmas morning. (7yr olds are not known for their patience.) Everything seems to be going along fine, but I wanted to add a new educational app to her iPod, and realized there's no App Store or iTunes. I'm guessing it's because she's 7 and that apple automatically restricts that, ok, fine, but without that, how can I even add anything from Family Sharing?

I figured maybe it was something I'd have to do through iTunes...So I sync up her iPod with iTunes on my laptop under her apple ID, and when I click on apps on the left hand side it shows all the apps with the Install or Remove button respectively, but they're grayed out and won't let me click on them. So I can't install or remove anything from there either. I go back to the iPod to try to delete an app by holding it down till everything is "shakey" and then none of the apps has the little x on it to delete. UGH.

I go under settings to try to find an option there, everything I try asks for a passcode, which apparently isn't the unlock code for the phone, and which I can't for the life of me remember setting up. I've tried every 4 digit combo I would think I would have used. So now I can't add/delete anything, music, apps, nothing. 😟

So obviously I have several questions:

1) Is it normal that there is no iTunes or App Store? I though they would be there but that she'd have to "ask to buy."

2) If they are supposed to be there, how do I get them back?
3) If they're not supposed to be there, how do I add or delete anything from the iPod? What's up with the "shakey" but no x thing?

4) What's going on in iTunes that I can't Install/Delete from there? I don't remember that being a problem when I initially set up the iPod.

Thanks in advance for any help you can offer.