How do I remove songs from an ipod touch (that we never put on there)?

Question

My daughter has a 5th gen ipod touch. When she opened it tonight, there are songs on there that we never put on it. I plugged her ipod in and opened itunes and the songs showing on the ipod are not showing up as being on the ipod when we are looking at itunes. In other words, when I am in itunes on my pc, when I click on music under her ipod's name it only shows a few songs (the ones we manually added) but not the ones showing up in the music app on the device. I have it set to manually manage music and always have. How do we get them off the ipod? I have home sharing turned off in the ipod and cloud is not enabled. How did they get on there and how do I get them off? These songs were not all bought through itunes but they are in itunes on the pc.
Also, "show all music" in the ipod settings is not on. Also, these songs were not on it yesterday. She listens to music every day using the shuffle feature and up until tonight, only the few songs she manually added would play. Now they all want to play (and it's not all of the songs from itunes either, so I'm not sure what made just these songs show up).

Answer 1

automatic downloads is turned off and already was. Nothing happens when I swipe left and right on songs that are not supposed to be on it (and they are also not showing as being on the ipod when it's plugged in). I don't think the songs are actually on the device but they do play when she shuffles her songs and she doesn't want that. And there's no rhyme or reason about which songs are showing in the songs list.
As for syncing, the ipod had not even been synced since we first bought it and put songs on it. The correct songs were on it the entire time until yesterday. It had not been synced at all (only plugged in to charge and itunes was never turned on while it was plugged in). What is involved in "syncing correctly"? Don't you just hit the sync button in itunes? We have 5 ipod touches in our household and have since 2009 or so, so we aren't complete beginners when it comes to them. We've synced before.
eta: for now I've created a playlist of the songs she originally wanted on there, so that she can at least shuffle through just those songs. We'd still like to solve this though.