If you can't place the Fetch app anywhere else in the Dock, except at the end near the trash, then it's not an application, and I need to know what you see. That's why i asked about what that shows in Get Info, after you select Fetch.
Does anything change (can you do what you need to do with the dock?) if you restart your Mac?
Another different tip that may help is to download and install your combined OS X updater. Here's the download page for the correct 10.4.11 combined updater.
http://support.apple.com/kb/DL170
I suggest that you be sure that you have a fresh restart before you install that updater, as it will "touch" a large part of your system as part of its job.
ftp client?
(not necessarily in this order...)
Fetch (you already know about that.)
Transmit (you need version 3 to use on 10.4
http://panic.com/museum/transmit/older/ )
Cyberduck (free!) (doesn't support 10.4, but you can download vers 3.2.1, which DOES support 10.4)
http://trac.cyberduck.ch/wiki/help/en/faq
or google for Mac ftp clients for OS X 10.4
Suggestion - before trying a different client, do 2 steps. Download and install the 10.4.11 combined updater, which I already mentioned. After that all completes, open /Applications/Utilities. Run Disk Utilities. Select your hard drive, and click the Repair Disk Permissions button. That permissions repair on 10.4 only takes a few minutes. Next step - restart.
Then - try your Fetch. Don't use the Dock this time for that, but go straight to the Fetch app, see if that opens successfully for you now...