I am not familiar with your web editor but a good place to start is to FTP the ENTIRE site to your local hard drive. Your web editor should have a much better chance at displaying the pages properly then. As the images are probably referenced using relative urls, you probably will not see them in your editor unless they exist in the same relative position on the hard drive. Similarly make sure you also have the CSS files in the proper relative position. 

"Design files" should not be necessary, at least not to make minor changes. First, what is the format of what's on the server now? Are they .html/.htm, .php, .asp, .jsp, or something else? Any of those can be edited with a plain text editor (e.g., ViM or Notepad++ on your PC, or on your server control panel > file manager > edit).

If the previous guy took off with the "design files" (whatever page/site editor he used), you'll have no choice but to edit by hand. You should be able to find text and such in the files and change them. If these pages were built using some kind of page/site editor, it may be difficult to find your way around. If the files you have at hand are .html or .htm, you may be able to import them into the page editor of your choice. If they're a scripted language (.php, .asp, .jsp, etc.) you would need a specialized editor to view them on your PC. What you might want to do is set up a "hidden" website on your server and upload to that to test changes. It could just be something like /home/ACCOUNT/public_html/dev/, or a subdomain "dev.yoursite.com". In the former case, if your pages use absolute paths (starting with "/"), you'd have to adjust those paths to add /dev in front. In the latter case, it would be automatic.