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JS, 303, 288, 275, 273, 274 (bug fix)

function z(s){return s.split('').map(function(d){if((r=Math.random)(b='qwertyuiop0asdfghjkl000zxcvbnm')>.1)return d
if((c=r(x=(d>'`'?b:(b=b.toUpperCase())).indexOf(d))*(/[a-z]/i.test(d)+2))<2)return c>1?d+d:''
for(a=0;!a;)a=b[x+[11,-11,1,-1][~~(r()*4)]]
return a}).join('')}

Algorithm for the keyslip:

• find char in string (or uppercase)
• add 11, -11, 1 or -1 to the index.
• if it's invalid (0 or null), re-roll

Non-Golfed version per request:

function z (s) {
  return s.split('') // split input string into characters.
          .map( // and then for each character...
    function (d) {
      r = Math.random; // set up a shortened form of Math.random
      // declare keyboard here because we have parens and we can save a delimeter.
      b = 'qwertyuiop0asdfghjkl000zxcvbnm';  
      if (r() > .1) {  // normal case
        return d;
      }
      numOptions = /[a-z]/i.test(d) + 2; // if it's a character, 1+2, else 0+2 options

      // test here because we have parens. x might be -1
      if (d > '`') { 
        x = b.search(d);  // x marks the spot
      } else {
        b = b.toUpperCase();
        x = b.search(d);
      }

      c = r() * numOptions; // chars can be 0-3, non-chars: 0-2
      if (c < 2) {                // this case is simple, so it comes first
        return c>1 ? d + d : ''; // double or omit.
      }

      // we must be in keyslip mode.

      // in the golfed code, this while loop become for loops, 
      // but it's really a while.
      a = 0;
      while (!a) { // that is, a != null && a != 0
        v = ~~(r() * 4); // 0, 1, 2, or 3
        newX = x + [11, -11, 1, -1][v]; // choose one
        a = b[newX];  // slip the key
      }
      return a;
    }
  ) // end the map function
  .join('') // and then reassemble the string
}