Once I publish an app at shyniapps.io or Shiny server so that users can access the app in the internet, can the users download the apps to their PCs and run the app from the PC?
No, they can't. If you want to make this possible, you can include a link to the app's code somewhere in the app itself.
Another option is Showcase Mode, which shows your app's code as part of the app so people can how it works (and copy and paste if they wish). More here:
The R Program that I am going to host in shinyapps.io or Shiny server calls other programs. These programs are opensource software used in proteomics research. The shiny application on the web will interact with the RStudio server program. Is that correct?
No, they can’t. If you want to make this possible, you can include a link to the app’s code somewhere in the app itself.
Could you show me how to do this?