How can I prepare a compiled Standalone Application such that the user can open it either in a Matlab command window or a web browser?
Running a compiled app by typing the name of the executable in the MATLAB Command Prompt is not supported (though you could call it using the system function.) For this use case I'd consider providing the non-compiled MATLAB program files (your scripts, functions, and/or classes) to users who need to run this in MATLAB.
How can I prepare the compiled app such that the user can pass the input arguments of the Matlab function and receive the output arguments?
See this documentation page. Input arguments should work (with the caveat that they are received by the application as char vectors by default, as noted on that page.) Where would you expect to receive any output arguments to if that were supported and you were (for example) running your application as a web app? That page talks about other ways to get information out of your application.
