Solution:
By the sounds of it, your custom theme is missing the common hooks that allow plugins to alter/output their code.
To take a simple example, every theme should have a call to wp_head() somewhere in the <head> section of the output page. This allows a plugin to “hook” into your <head>, and for example, output code to load its Javascript.
Here’s a real-life example. The WordPress Twentyeleven theme has this in its header.php file (traditionally the part of a theme that outputs the <head> section of any page):
... other <head> stuff
wp_head();
?>
</head>
The WP Nivo Slider uses this code when it calls wp_enqueue_script, for example, in its wp-nivo-slider.php file. Behind the scenes, wp_enqueue_script uses the wp_head() hook in the Twentyeleven theme to output the requested Javascript include in the <head> section (via a slightly circuitous route that ends up in wp_print_head_scripts by default.)
So, basically, if a plugin works with a provided theme, but doesn’t work with your custom theme, your task is to find the hooks that are missing from your theme that the plugin is trying to use.
If you check the WordPress Theme Development documentation you’ll find a list of hooks that themes should include under the section “Plugin API hooks”. These are, specifically:
wp_headwp_footerwp_metacomment_form
The important ones for most plugins will be wp_head and wp_footer. This is where most Javascript gets included, either in the head or the footer section (just before the closing <body> tag.)
Most plugins like Javascript sliders, image galleries, etc., will simply add a new script or two in the <head> or footer section of the website, and perhaps include CSS files to style their content, again in the <head> section, so those two are generally the only hooks they need.
So, my initial advice would be to make sure your custom theme includes a call to wp_head() at the end of its <head> section (copy the code from that working theme you’ve got) and also a call to wp_footer(), just before the closing </body> tag. The chances are good that that’ll get most Javascript-based plugins working.