How can I programmatically invoke an onclick() event from a anchor tag while keeping the ‘this’ reference in the onclick function?

You need to apply the event handler in the context of that element:

var elem = document.getElementById("linkid");
if (typeof elem.onclick == "function") {
    elem.onclick.apply(elem);
}

Otherwise this would reference the context the above code is executed in.

The best way to solve this is to use Vanilla JS, but if you are already using jQuery, there´s a very easy solution:

<script type="text/javascript">
    function doOnClick() {
        $('#linkid').click();
    }
</script>
<a id="linkid" href="/testlocation" onclick="alert(this.href);">Testlink</a>

Tested in IE8-10, Chrome, Firefox.

To trigger an event you basically just call the event handler for that element. Slight change from your code.

var a = document.getElementById("element");
var evnt = a["onclick"];

if (typeof(evnt) == "function") {
    evnt.call(a);
}

Granted, OP stated very similarly that this didn't work, but it did for me. Based on the notes in my source, it seems it was implemented around the time, or after, OP's post. Perhaps it's more standard now.

document.getElementsByName('MyElementsName')[0].click();

In my case, my button didn't have an ID. If your element has an id, preferably use the following (untested).

document.getElementById('MyElementsId').click();

I originally tried this method and it didn't work. After Googling I came back and realized my element was by name, and didn't have an ID. Double check you're calling the right attribute.

Source: https://developer.mozilla.org/en-US/docs/Web/API/HTMLElement/click

$("#linkid").trigger("click");