Ajax Class Efficiency
A review of the JavaScript part:


ajaxCon is a class/constructor, so you should call it AjaxCon, though really I would think about finding an even better name :)
ajaxFailure -> You have a number of functions that have ajax in their name, since these functions are in the Ajax constructor you can drop the ajax part
this.callback = false; -> Don't declare callback at all and then use if(this.callback) this.callback(responseArray);
This could use some newlines:
$.ajax( $.extend(true, {}, this.ajaxSettings, {data:{'ajaxFunction':functionName, 'arguments':argumentArray}} ) );

and also 

this.ajaxSettings = { url: window.location.href, type: 'POST', cache: false, data: {ajaxRequest: true}, success: function (returnData) {
    ajaxCon.ajaxReturn(returnData)
}};

Also the location of this.ajaxSettings is odd, this belongs on top of that function