JavaScript append a select menu option to a div tag

Question 1

Use var options = document.getElementsByTagName("option") in conjunction with options[eSelect.selectedIndex].innerHTML;, like this:

window.onload = function() {
    var eSelect = document.getElementById('question1');
    var optOtherReason = document.getElementById('displayresponse');

    // the option elements
    var options = document.getElementsByTagName("option");

    eSelect.onchange = function() {
        if(eSelect.selectedIndex === 2) {
            optOtherReason.style.display = 'block';

            var li = document.createElement("li");

            // eSelect.selectedIndex is the index (call it `n`) of the option
            // get that `n`-th option element from `options`
            // get its `.innerHTML`
            // and set to `li.innerHTML`
            li.innerHTML = options[eSelect.selectedIndex].innerHTML;

            var ul = document.getElementById("appendedtext");
            ul.appendChild(li);
        } else {
            optOtherReason.style.display = 'none';
        } 
    } 
}

EDIT:

If you want to display every change:

in the ordered list- http://jsfiddle.net/GaurangTandon/6h9TT/3/
in the div - http://jsfiddle.net/GaurangTandon/6h9TT/4/

Question 2

This is how I would do it:

var select = document.getElementById('question1'),
    response = document.getElementById('displayresponse');
select.onchange = function() {
    var selectedOption = select.options[select.selectedIndex].innerHTML;
    response.innerHTML = selectedOption;
    response.style.display = 'inline';
}

Fiddle