If I store music in CloudDrive, can I play it in somehow?
No. Unless you use a specific third-party app that allows this (I don't know of any) then synchronisation of music from a computer to an iOS device with iCloud Drive is not possible. 

However, you may find that iTunes Wi-Fi syncing sufficiently solves the problem you describe of having to physically connect your iOS device to your computer in order to synchronise media.