Yes and no.
Purchased items are not technically in their own library, they are part of your library. The purchased items have not been downloaded yet.
Depending on the view you are in, how they appear. Assuming you start at Library then choose a view, in Album view there is the arrow at the top right side of the Album to download the whole album. In Song view they are the ones without the arrow in a circle on the right side of the song.
To stop playing them (assuming actively listening), pick another song. To stop playing them (wanting to not play streaming music) use Library->Downloaded-><Album/Song/Artist view>.
To remove a downloaded song from the iPod Touch, tap the 3 dots (...) and tap Remove, when the prompt comes up choose Remove Download.
As for seeing these on the iPod Touch in iTunes, that is a bit problematic. iTunes has been terrible at showing the music on a device for several years now. The downloaded songs are SUPPOSED to be the grey circle songs, but that is no longer the case. Half of my synced songs show up as a grey circle these days. The downloaded songs just don't show anywhere.
I downloaded several songs to my iPhone just now, and synced, and none appear on my device in iTunes. This has been a problem in the past for users as they run out of space on their device, and they want to remove the downloaded songs to make room and can't find them. The solution to this is to sync with iTunes, removing all synced music, then open the Music App on the device and remove all tracks there (the list is all the downloaded songs) then sync with iTunes and add back the synced music.
iTunes is getting worse for managing music as time moves forward. We thought that the removal of Tones and Apps from iTunes management would help iTunes manage music better, but it really has been stagnant since that update. Smart playlist support hasn't made any changes in several years, and apps like Podcasts and Books don't support them any more.
You are not thick here, it is iTunes that isn't being user friendly and hasn't been for many years now. This keeps me answering questions though...