So if you don't know what's in the folder then you have no access and you are a hacker. What you want to do is known as "slurping", where someone attempts to download an entire website. Some browsers have this feature, like iCab, which might be a better choice for you. iCab's download feature includes downloading all linked pages from a site (called "Grabber").
The DocumentRoot, which is what you want, is off-limits to FTP because the DiectoryIndex page is all you can access. Subfolders are still accessible if indexing is allowed by the server.
If it's true that it is your own server space, then you may not even have shell access. Then you are stuck with FTP, which is very minimal and hard to use. I just use the Transmit application:
http://Panic.com/Transmit/