The problem is that your recursive call does not pass the second argument.

Without passing it, each recursive call will just populate its own, new array. It does return that array to the caller, but the caller (making the recursive call) ignores that returned value, so all the work of the recursive call is for nothing.

So the easy fix is to change this:

} else {
    recursion(nE);

to this:

} else {
    recursion(nE, resultAry);

The problem is how you are passing the processing of array, if the value is an array then you are keep calling it causing an infinite loop

function flatten() {
    var flat = [];
    for (var i = 0; i < arguments.length; i++) {
        if (arguments[i] instanceof Array) {
            flat.push.apply(flat, flatten.apply(this, arguments[i]));
        } else {
            flat.push(arguments[i]);
        }
    }
    return flat;
}

Demo: Fiddle

Here's a more modern version:

function flatten(items) {
  const flat = [];

  items.forEach(item => {
    if (Array.isArray(item)) {
      flat.push(...flatten(item));
    } else {
      flat.push(item);
    }
  });

  return flat;
}

You are checking for the presence of key each time through the loop over the object's properties. So you are getting as many values as there are properties on the object. So:

function getValuesByKey(object, key) {
  var values = [];
  recursiveFx(object);
  function recursiveFx(object) {

    if (key in object) values.push(object[key]);
    ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

    for (var property in object) {
      if (object.hasOwnProperty(property)) {
        if (typeof object[property] == "object") {
         recursiveFx(object[property]);
        }
      }
    }
  }
  return values;
} 

Alternative: use JSON.stringify with replacer

Anyway, you can do this more easily with

function getValuesByKey(object, key) {
  var values = [];
  JSON.stringify(object, function(k, v) { 
    if (k === key) values.push(v);
    return v;
  });
  return values;
}

This uses the replacer parameter to JSON.stringify to intercept each key value pair. The stringified value itself we don't need and throw away.

You're creating a new array instead of passing it to the recursive call.

Do this instead.

DEMO: http://jsfiddle.net/kDtZn/

function addToArray(array) {
    array.push(prompt("Add items to array or 'q' to stop"));
    if (array[array.length-1] == 'q') {
        array.pop();
        document.write(array)
    }
    else {
        addToArray(array);
    }
 }
 addToArray([]);

Now you start with an empty array, and for each recursive call, it passes the same array forward.

Also, I changed it so that it doesn't use .pop() in the if() condition, otherwise you'll always end up with an empty array when it comes time to write it. (The .pop() method actually removes the last item.)

Finally, make sure you're not using document.write after the DOM is loaded. If so, you need to change it to use DOM manipulation methods instead.

You could take a different approach so that you don't need .pop() at all.

DEMO: http://jsfiddle.net/kDtZn/1/

function addToArray(array) {
    var item = prompt("Add items to array or 'q' to stop");
    if (item == 'q') {
        document.body.textContent = array;
    } else {
        array.push(item);
        addToArray(array);
    }
}
addToArray([]);

The reason your while loop didn't work is very likely because of the original .pop() issue.