AJAX, JQuery, JQuery UI, Stylesheet, Custom JS -- NOT LOADING ALL

Question 1

you are using two jquery and jquery ui. You must use one. since .min.js and .js are same. min.js is just the compressed version of .js

Ie the below code is enough:

<head>    
<link rel="stylesheet" type="text/css" href="CSS/nudgeme.css">
<link rel="stylesheet" href="//code.jquery.com/ui/1.10.4/themes/smoothness/jquery-ui.css">
<link href='http://fonts.googleapis.com/css?family=Cabin:400,600' rel='stylesheet'     type='text/css'>
<link rel="stylesheet" type="text/css" href="CSS/nudgeme.css">

<!-- First import jquery -->
<script src="//code.jquery.com/jquery-1.9.1.js"></script>

<!-- Then Jquery library imports -->
<script src="//code.jquery.com/ui/1.10.4/jquery-ui.js"></script>    
<script type="text/javascript" src='js/nudgeme.js'></script>

<!-- Then Custom script that uses the above libraries-->
<script type="text/javascript" src='js/custom.js'></script> 
</head>

Use custom.js below all other imports. because javascript executed from top to bottom. so for example if your script uses a javascript function(for eg: jquery $) and you write the code above the import of jquery.js it simple won't work.

Question 2

Ok i did not found <!DOCTYPE html>(so it is not html5) in your code and so i recommend you to mention type="text/javascript" in you script tag

<script type="text/javascript" src="//code.jquery.com/jquery-1.9.1.js"></script>
<script type="text/javascript" src="//code.jquery.com/ui/1.10.4/jquery-ui.js"></script>

for reference link