Import RSS feed from categories into external site

Sorry this post will be a bit long but I have a problem that is seriously driving me insane and I just can’t get round it. Please bear with me, as if anybody can help I will be truly truly grateful.

I am a new WordPress user. For a number of years my IT department had a system set up on another site (Article Manager if that means anything to you); but have now told me I have to change to WordPress, so am trying to find out how to do what I used to do in ArtMan with WP.

Without boring you with the details of why, I have a site where 50+ users have the potential to post into a number of specific categories; and I need to be able to export a feed of their posts (for each particular category) into our external website. Hope that makes sense.

HOW I USED TO DO IT ON ARTICLE MANAGER
I used a simple bit of Javascript code. On ArtMan, each feed corresponded to a Javascript URL [I’m not an IT professional so please don’t shoot me down if I get the technical terms a bit wrong] such as http://newbattle.mgfl.info/artman2/publish/syndicate_27.js. On the external webpage where I wanted the feed to appear I would then use code along the lines of this:

<!-- begin syndicated content -->
 <script language="Javascript1.1" type="text/javascript" ><!--
  var syndicate = new Object;
  syndicate.title_fontbold    = true
 [... 20 lines missed out for brevity ...]
 syndicate.not_found_message = '';  // -->
    </script>
  <script language="Javascript1.1" type="text/javascript" src="http://newbattle.mgfl.info/artman2/publish/syndicate_27.js"></script>

Like I say, this did the job nicely

WHAT I’M TRYING TO DO NOW
I need to try to replicate this functionality in WordPress. As far as I can work out, there is no *.js URL in WordPress, so I can’t use the same Javascript code.

From what I can gather the only way I can do it now is as an RSS feed. However I have yet to find any code that imports RSS from my WordPress site into my external site.

I have managed to discover that the RSS URL for an individual WordPress category for my site is along the lines of http://newbattle.mgfl.net/?cat=3&feed=rss2 [which in itself confused me as I was under the impression RSS feeds should end in *.rss]

However I cannot work out how to make this appear in my website. Using iframes hasn’t worked either as all the headings and stuff get in the way.

Does anybody have any idea at all of where I can go from here, as I am well and truly at a dead end. I had a nicely working system under Article Manager and am worried I will have to completely replan my site if I can’t get this to work.