Well, I'm not familiar with the component you're on about, but generally the source folder and it's subfolders will contain the source files you'd need if you decided that you wanted to modify/customise the component. You don't need to put any of these files onto your server.
The stuff in the deploy folder (and any sub-folders) however will need to be copied to your server. In order for the component to work the file/directory structure on your server would have to be identical to the directory structure here.
It seems strange that you can't create sub-folders on your server....Without the appropriate files in the appropriate subfolders, your component will not work. It's as simple as that!
All you should need to do is duplicate the folder structure for the deploy folder of your component on your server, copy all of the relevant files and then embed the .swf into your page either manually with object/embed tags or by using a script like swfobject.js (a cross browser compatible script for auto detecting browser versions and embedding flash in html pages in a manner compatible with the browser - google it!)
The only other way to do it would be to edit the sources of your component so that it looks for the xml files and images in the same folder as the .swf instead of looking in the subdirectories, rebuild the component and move all of the xml files up to the root folder and then upload everything to your server and embed the .swf in your page/pages....But that's just bloody stupid IMHO. You should be able to create as many subdirectories as you need...What is stopping you from creating the subdirectories on your server??
Cheers for now,
Jas.