Greetings, Mr Crun, sir.
I have a load of music on my PC, copied onto the iPod and the iPad. It's all classical, sorted into playlists as appropriate, and it plays movements in the correct order every time. Daughter 1 copied some of our choir onto a little MP3 player, and that behaved nicely.
You may be suffering from something in the library PC that isn't ripping the CDs correctly.
Does your Chromebook have a USB socket on it where you could connect an external CD player to start the ripping process? I have a nifty little external CD unit here that I rarely use - you're welcome to it if it'll help. PM me an address and it's yours.
Could you run something downloadable like iTunes on the Chromebook? I wouldn't use iTunes itself if you don't have to - it's bloated and very user-unfriendly but pretty much mandatory for iThings. There's a variety of software available for the purpose - I'd defer to mixture as to which is most suitable.