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Using jQuery, is is possible to watch a DOM element and report each time something is done to it?

I'm working on a site containing a confusing chain of asynchronous effects. Often, effects do/undo each other several times before stopping. I am having a very hard time following the spaghetti code.
Is it possible to set a callback to run any time a DOM element is manipulated so I can follow the chain of effects the code is applying to it?
Edit: I am currently adding dozens of console.logs everywhere and wanted a cleaner approach.

One approach is hooking into the core jQuery functions and log the information you want. Here's some very rough code on a few functions that i've hijacked i.e hide and animate.
var log_for_functions = 'hide animate'.split(' ');
$.each( log_for_functions , function(){
var function_name = this;
var original_function = $.fn[ original_function_name ]; // get a reference to old function name
$.fn[ function_name ] = function () {
var r = original_function.apply(this, arguments);
console.log( function_name + ' called on element ', this );
return r;
}
});
Now if you hide an element using jquery $('#test').hide() then you will get the following output.
hide called on element [context: #test, selector: "#test"]
You can change this to fit your needs and even hook into other functions.

Not without setInterval.
You could just have the code that applies an effect also note what changed.
You could do this with data attributes.
<div data-state="some state" ...
console.log(targetElement.getAttribute("data-state");
targetElement.setAttribute("data-state","changes");
MakeChanges(targetElement);

The following article introduces two techniques that might work in your case, Mutation Observers and Keyframes:
http://www.backalleycoder.com/2012/08/06/css-selector-listeners/
For each technique the article points to the related library.

Creating a small jQuery plugin

I created this plugin to make work around an application easier.
Here is the link : http://jsfiddle.net/X5Squ/
My problem it that it always uses just 1 of the elements, please don't edit the data and data5 functions as these work perfectly for other parts but I need my function called jtoggle to work.
Any help much appreciated! Thanks.

$(document).ready(function (){$('.jtoggle').jtoggle(true);});
Have you tried using .each on this? I think the issue is that it isn't passing an array of DOM elements. I lack much experience in creating plugins, but it seems this can be easily averted by doing the following:
$(document).ready(function (){
$('.jtoggle').each(function(){
$(this).jtoggle(true);
});
});
(Which would also mean that you can safely remove the .each you have in jtoggle itself)

In your plugin you should act on each matched element and then return all the matched elements in order to maintain chainability:
$.fn.jtoggle = function (addUnderline) {
return this.each(function () {
// Do what you need on this matched element
});
};
Maintaining chainability means we can do stuff like:
$(".jtoggle").jtoggle(true).addClass("xyz");

Most efficient way to re-use jQuery-selected elements

I can imagine the correct answer to this based on theory, but I'm just looking for some confirmation. I'm wondering what the most efficient way to re-use a jQuery-selected element is. For example:
$('#my_div').css('background','red');
//some other code
$('#my_div').attr('name','Red Div');
vs.
myDiv = $('#my_div');
myDiv.css('background','red');
//some other code
myDiv.attr('name','Red Div');
I assume the second example is more efficient because the element #my_div doesn't have to get found more than once. Is that correct?
Similarly, is it more efficient to first save $(this) in a varaible, such as 'obj', and then reuse 'obj' rather than using $(this) over and over? In this case, jQuery isn't being forced to find an element over and over again, but it IS being forced to convert this to a jQuery object [$(this)]. So as a general rule of thumb, should a jQuery object ALWAYS be stored in a variable if it will be used more than once?

You should write your code such that you limit the number of DOM traversals.
When you write something like this:
$('#my_div').css('background','red');
//some other code
$('#my_div').attr('name','Red Div');
You are finding #my_div twice, which is inefficient.
You can improve this either by assigning the result of a selector (i.e. var x = $('.something')) and manipulate the variable x, or you can chain your method calls like this:
$('#my_div').css('background','red').attr('name','Red Div');
You'll see the above code used a lot, because you're finding the element once. The css() method will apply a CSS style and return the actual result of $('#my_div'), so you can invoke another method, in this case attr().
My preferred way of handling the re-use of selectors is to store them as a variable, and wrap my stuff in a closure.

if you're using jQuery selector (like $('#element')), then yes, you should always store your results.
if you're using object and wrapping it in jQuery (like $(this)), it's not necessary, because jQuery doesn't need to search for that element again.

One thing that I find is generally overlooked is just how powerful jQuery chains are. It may not be so noticeable, but since jQuery caches your wrapped elements within a chain, you can modify elements, go into a more specific subset, modify, then go back up into a a general superset without much overhead.
I expect something like (pardon the example)
$('#myDiv')
.addClass('processing')
.find('#myInput')
.hide('slow')
.end()
.removeClass('processing')
;
to be better performance-wise than even
var $myDiv = $('#myDiv').addClass('processing');
var $myInput = $('#myDiv #myInput').hide('slow');
$myDiv.removeClass('processing');

This also holds for applying the jQuery function to elements returned in an event handler. Try to avoid applying $(...) too many times, because this is slow. Instead create a variable that contains the result of $(...). Good practice is to start the variable with a $, which gives a hint about the jQuery object inside the variable.
$('a').click(function(){
var $this = $(this);
$this.addClass("clicked");
$this.attr("clicked", true);
});

Wrapping a DOM element inside a JavaScript object

I've noticed a common pattern in the JavaScript I've been writing and was wondering if there is already a pattern out there that defines something similar as best practice? Essentially, it's how to get a DOM element and wrap it inside / associate it with a JavaScript object. Take this example, where you need a filter in your web app. Your page looks like this:
<html>
<head></head>
<body>
<div id="filter"></div>
</body>
</html>
You'd then wrap the element like so:
var myFilter = new Filter({
elem: document.getElementById('filter'),
prop: 'stacks-test',
someCallback: function() {
// specify a callback
}
});
And the JavaScript (where spec is an object passed to the constructor):
var Filter = function(spec) {
this.elem = spec.elem;
this.prop = spec.prop;
this.callback = spec.someCallback;
this.bindEvents();
};
Filter.prototype.bindEvents = function() {
var self = this;
$(this.elem).click(function(e) {
self.updateFeed();
};
};
Filter.prototype.updateFeed = function() {
this.prop; // 'stacks-test'
this.callback();
// ...
// code to interact with other JavaScript objects
// who in turn, update the document
};
What is this kind of approach called, and what are the best practices and caveats?

You might be interested in Dojo's widget library, Dijit - if I'm understanding your question correctly, it essentially does what you're asking, and a whole lot more.
In Dijit, a widget essentially encapsulates a DOM node, its contents, any JavaScript that defines its behavior, and (imported separately) CSS to style its appearance.
Widgets have their own lifecycle, registry, and events (including many which simply map to DOM events on a node within the widget, e.g. myWidget.onClick could effectively call myWidget.domNode.onclick).
Widgets can (but don't have to) have their initial contents defined in a separate HTML template file, through which it's also possible to bind events on nodes within the template to widget methods, as well as set properties on the widget that reference particular nodes in the template.
I'm barely scratching the surface here. If you want to read more on this, you can start with these reference pages:
http://dojotoolkit.org/reference-guide/dijit/info.html
http://dojotoolkit.org/reference-guide/dijit/_Widget.html (the base that all widgets extend)
http://dojotoolkit.org/reference-guide/dijit/_Templated.html (RE the HTML templating)
http://dojotoolkit.org/reference-guide/quickstart/writingWidgets.html (useful information when starting to write your own widgets)
http://dojotoolkit.org/reference-guide/dijit/ (for a bunch more info)
All said, I don't know what you're ultimately aiming for, and maybe this is all a bit much for your purposes (considering I'm throwing an entire other library at you), but figured it might pique your interest at least.

Continuing from my comment on the question, jQuery is a potential tool for the job, as it already provides some of the foundations for what you're after. However, having said that, it does introduce complexities of its own, and further, not all "jQuery ways" are equal. I'll suggest one way of using jQuery as your "object model", but it may or may not suit your needs.
First things first. The philosophy of jQuery is that you start everything by selecting the element first, using $(), or equivalently jQuery(). All operations conceptually begin with this. This is a slightly different way of thinking compared to creating an object that wraps an element and keeping a reference to that wrapper, but essentially this is what jQuery does for you. A call to $('#some-id') grabs the element with id of "some-id" and wraps it in a jQuery object.
One way: Write "Filter" plugins.
Replace your constructor with a initFilter() jQuery method. You can do this by modifying the jQuery prototype and using the jQuery object as your wrapper. jQuery's prototype is referenced by jQuery.fn, so:
jQuery.fn.initFilter = function (prop, callback) {
// Save prop and callback
this.data('filter-prop', prop);
this.data('filter-callback', callback);
// Bind events (makes sense to do this during init)
this.click(function () {
$(this).updateFeed();
});
};
Then do a similar thing for updateFeed():
jQuery.fn.updateFeed = function () {
this.data('filter-prop');
this.data('filter-callback')();
});
And use it like this:
$('#filter').initFilter(prop, callback);
Note that updateFeed can simply be in-lined into the click handler to prevent unnecessary pollution of the jQuery namespace. However, one advantage of using jQuery like this is that you do not need to keep a reference to the object if you need to invoke some function on it later, since jQuery ties all references to actual elements. If you'd like to call updateFeed programmatically, then:
$('#filter').updateFeed();
will then be invoked on the correct object.
Some things to consider
There are certainly downsides to this method. One is that all properties, which we've saved against the element using .data(), are shared between all jQuery functions that act on that element. I've attempted to alleviate this by prefixing the property names with "filter-", but depending on the complexity of your object(s), this may not be suitable.
Further, this exact method may not be so suitable for objects that require a lot of manipulation (i.e. objects with many functions) since all of these functions become common to all jQuery objects. There are ways to encapsulate all this which I won't go into here, but jQuery-ui does this with their widgets, and I'm experimenting with yet another alternative in a library I'm creating.
However, pulling back a bit, the only reason I suggested using jQuery in the first place is that your Filter object appears to be heavily tied to the DOM. It binds events to the DOM, it modifies the DOM based on user interaction, basically it appears to live in the DOM, so use something DOM-based, i.e. jQuery.

Using jQuery in a JavaScript function

function divlightbox(val)
{
if(val)
{
val=val.replace( /^\s+/g, "" );
var count_js=0;
var big_string='';
document.getElementById("video_lightbox").innerHTML="";
document.getElementById("divlightbox").style.display = "block";
$("#video_lightbox").css({"height":"430px","top":"10%","width":"480px"});
I found out that the error is in the above. My question is can't I use jQuery and traditional JavaScript at same time? I have done coding like this numerous times and never ran into a problem like this. I used to use jQuery methods like .hide() and .css() inside JavaScript functions but this time it doesn't work.
Thanks in advance.

While the other answers fix the specific problems, I don't think the OP's question (in bold) is really answered here, as depending on the specific context, $ may possibly not be defined as a jQuery object yet (having had this problem myself a few times now.)
In which case you would need to do something like:
function divlightbox(val) {
// ...
// just use jQuery instead of $ one time
jQuery("#video_lightbox").css({"height":"430px","top":"10%","width":"480px"});
}
OR
function divlightbox(val) {
// define the $ as jQuery for multiple uses
jQuery(function($) {
// ...
$("#video_lightbox").css("height":"430px");
$("#video_lightbox").css("top":"10%");
$("#video_lightbox").css("width":"480px");
});
}

jQuery is JavaScript so YES. Instead .innerHTML="" just use .empty(). Instead .getElementById() use $('#..') and so on.

to do things like hide(); and css() you need jquery objects. you can't do them to dom elements.
so you could do $('#video_lightbox').html("");
or
$('#video_lightbox').empty();

You must provide error in javascript console.
1) Do you pass a val argument to divlightbox function()? When do you call it?
2) why do you use the same identifier divlightbox both for a function and for a div id? Change name to the function please, maybe the problem could be here.
3) Always check if video_lightbox and divlightbox exist before accessing them.