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deleted 15 characters in body

##JS, 303##

function z(s){return s.split('').map(function(d){if((r=Math.random)()>.1)return d
if((c=r()*(/[a-z]/i.test(d)?3:2))<2)return c>1?d+d:''
if((x=(b="qwertyuiop0asdfghjkl000zxcvbnm").indexOf(d))<0)x=(b=b.toUpperCase()).indexOf(d)
for(a=0,n=11;i=!a;a=b[x+n])for(v=r()*4;v>i;n=i++%2?-n:1);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
      if (r() > .1) {  // normal case
        return d;
      }
      numOptions = /[a-z]/i.test(d) ? 3 : 2; // if it's a character, three options
      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.
      b = "qwertyuiop0asdfghjkl000zxcvbnm"; // keyboard map
      x = b.indexOf(d);  // x marks the spot
      if (x < 0) {  // not here? must be a capital letter!
        b = b.toUpperCase();
        x = b.indexOf(d);
      }

      // in the golfed code, these while loops become for loops, 
      // but it's really a while.
      a = 0, n = 11, i = 1;
      while (!a) { // that is, a != null && a != 0
        i = 1;
        v = r() * 4; // so v is a random number <= 4
        while (v > i) { 
          n = i++ % 2 ? -n : 1; // n steps in this pattern: [11, -11, 1, -1]
        }
        a = b[x+n];  // slip the key
      }
      return a;
    }
  ) // end the map function
  .join('') // and then reassemble the string
}