Can Checksum alerts make it to the end user?
Yes, any corruption alerts can make it to the end user.

To see it yourself, grab one of the corrupt databases from Steve Stedman's Corruption Challenge. Attach the corrupt database to one of your development environments, and run the query that Steve describes for that database. You'll see corruption errors - just as your end users would.

Your application (C#, Java, whatever) can be coded to trap incoming errors and display different things to the users, but that's an exercise left for the reader.