Will my Shiny app work properly on a mobile device (iPhone, Android, iPad, Kindle, etc.)?

All shiny applications are HTML/CSS and Javascript. Shiny’s default components make heavy use of Bootstrap which supports a responsive web design. You should find that most applications you create will run properly on an iPhone, Android, or iPad without modification, however if you need to tweak things, you have full control to do that by dropping down to the javascript and html layers.