Quizzes are popular. Anyone on Facebook notices those Buzzfeed quizzes being shared all the time.
jQuery is the library you want to use. It's the core script you'll need.
Then jQueryUI is an addon you can use (http://jqueryui.com/). It's great for sliders and such. All cross browser compatible.
If you're looking for responsiveness, Bootstrap is awesome. (http://getbootstrap.com/) - you can make the right floating column with the affix component.
Markup and style your quiz, and when you're done you have two options: PHP Form or AJAX request. If you're taking the "damn this is a sexy and interactive survey" route, Ajax is the route you'll want to take. jQuery has decent documentation on Ajax requests. Simply request a PHP file from your ajax request with the variables you want to send to PHP, and your PHP script can then do the math, query a database or just return a basic message.