Are you sure it's free?

Yep, it's completely free... to include making the source code available as "public domain" There is no licensing of any kind, and no restrictions on its use (to include making money from it!)

I get a message that says that I should install something

The WRPN program require the ".Net Framework" to be installed on the PC. Generally, you should install the latest version.

Download and install the latest .Net Framework from: http://www.microsoft.com/net/

The help file comes up but just contains an error message

Microsoft recently changed the security settings for compiled HTML help files (*.chm). These new restrictions apply when the help file is being open from a network share. See the following for a fix: http://support.microsoft.com/kb/892675

Where can I get the User's Manual for the real calculator?

I'm working the HP to get permission to reproduce the original manual as a PDF file. In the meantime, you might be able to find a copy elsewhere on the web. Try the this google search link: https://www.google.com/search?q=hp16c+users+manual

What about advertisement and user privacy?

This application does not have any advertisement and does not collect any data at all! The application uses the following Android permissions: