That doesn't depend on shiny-server but on the computational resources available in the physical server (or the resources made available to you in a cloud computing service if that is your case).
A server can run more than one service and for running shiny-server you need to have R installed in the system but your question is not clear. Can you clarify?
If your application requires more computational power, then yes, you need to "increase" your server but that is a simplistic view of the problem, there are many factors to consider when scaling up a deployment.
Possibly, but we can't give you any specific advice without knowing the specifics of your app and your deployment setup.