I got this hash RIPEMD160(TenFourFox7450-FPR19.app.zip)= 4b96b198ac147c6c6d6520f7eff32a9b8e671a8c. Is it not ideal to download the zip file to a usb drive then move the zip file from the usb drive to the Applications folder on my mac?
If it's Archive Utility (check the window titlebar while it's unpacking), then you're doing everything right, and I don't have any other suggestions for you as to why it's not working properly on your machine.
This is getting out of scope for TenFourFox specifically, but to answer your question your iBook G4 will not run 10.6 or later, so to run TenFourFox your choices are 10.4 and 10.5. In general 10.4 will be faster, but 10.5 has more software options.