JQuery Cycle Plugin and other JQuery issues

I’m using the iNove theme and I’ve heavily modified the theme to the point where it no longer looks like the original. I’m trying to get various JQuery functions working, but I’m having some difficulty.

I created a simple widget with a slideshow that uses JQuery. In functions.php I did include JQuery:

if ( !is_admin() ) {
   wp_deregister_script('jquery');
   wp_register_script('jquery', ("https://ajax.googleapis.com/ajax/libs/jquery/1.3.1/jquery.min.js"), false);
   wp_enqueue_script('jquery');
}

However it doesn’t work. I add the script to my header.php:
<script type="text/javascript" src="<?php bloginfo('template_url'); ?>/js/jquery-1.3.1.min.js"></script>

And it works… I thought that the code in functions.php would be added to my header. Am I doing something wrong?

I’m also trying to get the JQuery Cycle Plugin to work, but I can’t seem to get even the demo working in wordpress.

https://jquery.malsup.com/cycle/

Basic demo: https://jquery.malsup.com/cycle/basic.html

I think this has to do something with the functions.php. Can anyone help me figure these issues out?