Additional SQL Server features and topics not covered by specific categories
Using SSMS without SQL Server is pretty pointless, but you don't need to have SQL Server installed locally. If you are in a corporate environment, there may be SQL Server instances on other machines. You can also connect to various services with SQL Server in the cloud.
Why your installation gets stuck I don't know. I would recommend that you select the download option and then install from the ISO. That gives a little better visibility of what is going on.
Windows 10 is certainly supported for SQL 2022.
The log files as SeeYa listed can be help to understand what is going on.