"Live migration" refers to how Compute Engine transparently migrates a VM to another physical host [1]. Disk and memory is copied over, and they have some ridiculous technology that keeps network connections alive and re-attaches them to the new VM when it's been switched over, so that it causes, in principle, zero disruptions. This is much more magical than other providers, such as AWS and DigitalOcean, where such a migration results in a reboot.

You can run PostgreSQL on a VM just fine. You just have to manage itself. Cloud SQL comes with some upsides (zero management, spectacular HA failover capabilities) and some downsides (lack of extensions, lives on a separate network, no control over maintenance window); you have to decide what you're willing to live with.

You can set the upgrade window, but it can't be predicted. What you can control is the order — e.g. set your staging instance to "early" and production instance to "late", then hopefully staging should be upgraded first and you'll know ahead of the production upgrade if any issues arose.

[1] https://cloud.google.com/compute/docs/instances/live-migrati...