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);
    }
  )
}

Direct answer

The following formula return the expected results for the three cases exposed in the question. It should be added to cell A2 and cells in A3:A13 should be empty.

=ArrayFormula(
  IF(
    NOT(
      (ROW(B2:B13)-1=1)
      *
      (B2:B13="$")
    )
    *
    (
      (B2:B13=">")
      +
      (B2:B13="$")
      +
      (ROW(B2:B13)-1>=Vlookup("$",{B2:B12,ROW(B2:B12)},2,0))
    )
    *
    NOT((B1:B12=">")
    *(B2:B13="$")),
    ".",
    " "
  )
)

Explanation

In this case the use of ArrayFormula function together with references to B2:B13 will return an array of 12 rows and 1 columns with the expected values.

The comparison of a range or array with a scalar value returns an array of the same size of the range or array. All the tests should return an array of the same size.

AND and OR functions could not work with arrays, but Google Sheets interpret TRUE as 1 and FALSE as 0, so, instead of using AND function use * and instead of OR use +.

As + and * take precedence over comparison operators each test should be enclosed in parentheses.