I installed the free version of the Download Manager plugin and did run into several error notices during the activation and usage, but they were all minor in nature and did not break the functionality of the Promotion Slider plugin.
Upon closer inspection, it appears that the real issue is with your theme. Your theme is loading its own version of jQuery, which is version 1.4.2. However, WordPress currently ships with version 1.11.0. The download manager plugin uses functionality from jQuery version 1.7+, which is why everything starts working again once the plugin is disabled. To verify this, leaving all plugins active and switching to one of the default WordPress themes should prove that the Download Manager and Promotion Slider plugins can get along just fine.
Beyond the jQuery issue, it seems that your theme is actually loading the scripts and stylesheets after the opening body tag rather than before the closing head tag.
I’m not sure if you or someone you’ve hired is the author of the theme, but it seems that the theme is going to need some work. If the theme is one you purchased, you may want to consider switching to another theme.