Overview

The OP basically wants to be able to use the spreadsheet functions over the original data without having to scroll across a large number of rows and columns. His background make him think about having custom views and a easy way to change from one to another but Google Sheets doesn't have a built in feature to manage custom views.

Fortunately the fourth requirements are doable by using built-in features and Google Apps Script

Rows could be hidden by using filters but columns only could be hidden manually or by using code.

For 1, and 2 use Google Apps Script to create custom menus that apply the hide/show columns and optionally rows over the "master sheet". This will allow the user to edit the data directly over the "master sheet" and sort it.

Code example

The following code creates a menu for handling the hide/show operations of columns specified in a global variable.

var cols = [1,5,7]; //columns to be hidden/shown

function onOpen() {
  var ss = SpreadsheetApp.getActiveSpreadsheet();
  var menuEntries = [ 
    {name: "Hide", functionName: "hideCols"},
    {name: "Show", functionName: "showCols"}
    ];
  ss.addMenu("Custom Views", menuEntries);
}

function hideCols() {
  var sheet = SpreadsheetApp.getActiveSheet();
  cols.forEach( function(col) {
      sheet.hideColumns(col);
    }
  )
}

function showCols() {
  var sheet = SpreadsheetApp.getActiveSheet();
  cols.forEach( function(col) {
      sheet.showColumns(col);
    }
  )
}

If your data is in A1:D10, maybe:

=filter(A1:D10,A1:A10>500)

To allow for "new ones will be added" put the formula outside A:D and adjust to:

=filter(A1:D,A1:A>500)  

Edit to include comment from someone who knows (@AdamL):

it is not possible to do what you want with the Sheets filter tool alone. You can combine both the filter tool and a FILTER or QUERY formula (eg apply the filter tool to the output of pnuts' solution, which could also reference the 500 part in another cell, perhaps in the header row to the side). Alternatively, a helper column(s) could be used to simulate the required functionality.