On-Premise SQL Server and SQL Server on a VM hosted in Azure are roughly equivalent and can be configured to be part of the same availability group, although as you mentioned the failover is manual rather than automatic.
This link https://learn.microsoft.com/en-us/azure/virtual-machines/windows/sqlclassic/virtual-machines-windows-classic-sql-onprem-availability provides more details.
Azure Database (As A Service) is a different thing. It is definitely based on SQL Server and it's High Availability is provided by something that is very similar to Always On Availability Groups. But it's completely managed by Microsoft so you won't be able to add it as a member to your own Availability Group.
This link https://learn.microsoft.com/en-us/azure/sql-database/sql-database-high-availability has more details.
To answer your second question you should view resources in different regions as if they are in separate data centers (which they are). You can make automatic failover work in these situations by using VPN tunnels and judicious use of cloud witnesses. However, I would advise against failing between data centers automatically, that should be a deliberate decision.
But mostly this depends on your HA/DR requirements. I love Azure and we use it heavily in my current organization, but stretching across datacenters or on-prem/cloud is not as pain-free as Microsoft would have you believe.
