Will I be notified that BACKUP encountered a CHECKSUM error if I use CONTINUE_AFTER_ERROR?
Yes it will continue and finish, there is an example in this older blogpost. It also contains a broken database file (for older versions) to play with:

https://blogs.msdn.microsoft.com/sqlserverstorageengine/2007/04/17/example-corrupt-database-to-play-with-and-some-backuprestore-things-to-try/

I think it is actually quite important as a dba to simulate all those cases, I know it from experience with customers it is risky and time consuming if you have to research those methods in a downtime. So kudos for asking. Here is a lunatic/ingenious method to do corruption with SQL: 

https://sqlblogcasts.com/blogs/tonyrogerson/archive/2007/03/10/how-to-create-a-corrupt-database-using-bulk-insert-update-and-bcp-sql-server-as-a-hex-editor.aspx