Android supports cloud and local backup of APK files, application data, and so on. Is the device keystore included in the backup?
(I don't think you can extract private keys from an hardware-backed keystore even in rooted devices, not so sure about the software-based version used when there is no hardware support.)
No. The whole point behind the hardware-backed keystore is that it's resistant to even privileged attacks with root access.
For this reason, not even the OS can access raw key material from the hardware-backed keystore.
While it would be theoretically possible for the OS to do a keystore backup on lower end devices with a software-backed keystore, this is simply not done or allowed by the OS for security reasons.