Bootstrap Carousel-inner "autoplay"

I've just set up a Fiddle containing the carousel of your example, jQuery version 2.1.3, bootstrap.js and bootstrap.css as external resources and your script to initalize the carousel

$('#myCarousel').carousel({
  interval: 3000,
  cycle: true
}); 

and it's working - it starts when the page loads and the controllers are working, too. That it's not working for you can be an issue of loading bootstrap before jQuery - jQuery has to be included before bootstrap. To check if this is the case, see if there are any error / console message like "Error: Bootstrap's JavaScript requires jQuery". Also you're including 2 different jQuery versions - 1.11.1 and 2.1.3. If not necessary, e.g. for some plugins relying on an older jQuery version, try just to load the latest one.
In addition, you're placing script tags between the closing body and html tags which can work, but isn't valid - see link, script tags between </body> and </html> for details.
As reference for using different jQuery versions in case this would really be necessary, you can check http://api.jquery.com/jquery.noconflict/

You could do it through the HTML with the data-interval tag. Insert it into <div id="myCarousel" class="carousel slide"> so it looks like this:

<div id="myCarousel" class="carousel slide" data-interval="3000">

The value of 3000 can be adjusted to a higher number to decrease speed and a lower number to increase speed.