I'm not familiar with rhandsometable. Do you mean rhandsontable? That would be for editing the data in the UI, which is more the 2nd / 3rd bullet. To get data into the UI, you will either be pulling it from a source other than the UI (database, file on the operating system running the R process, or user-provided file through an "Upload). Then to get data out of the UI, you will be pushing it to a source other than the UI (database, file on the operating system that is running the R process via one of the save or write methods, or user-provided file through a "Download"). You do have to think a bit differently than just your "normal" R session, though, because the interface is a web browser instead of an R console.
It sounds like you need to do a bit more thinking about what exactly you are trying to set up, but it sounds like you definitely have the tools to do so. Note that there are security implications of all of these processes if you expose the app to users, so it is good to keep that in mind.