Yes, there is an easy way to update, install and uninstall programs under Mandrake.
It's called Urpmi, it's considered one of the best and easiest package management utlilities for Linux (second only to Debian's APT), and you can access it from the Mandrake Control Center=>Software Management. From here, you can install programs from the huge Mandrake repositories (meaning that you need not trawl the Internet for Mandrake-specific RPMs) and uninstall previously installed programs that you don't want. You can also use RPM from the command line for individual programs, but this can be an annoying procedure, if the program has dependencies (which are automatically downloaded and installed by Urpmi, but have to be installed by hand from the command line).
Your problem is that the Software Sources Manager doesn't know where to find any of the installable programs except for on the CDs you installed from, but that is easily fixed.
Go to Easy URPMI
and follow the instructions to add remote (Internet) sources to your sources list. Then you can fire up the Install Software tool. Change the filter from the default "Mandrake Choices" to "All programs by ...." (you can choose whatever sorting mechanism you like, alphabetical, program type grouping, source repository). Then click on whatever programs you like and start installing.
You will get some messages when the actual installs begin, telling you that there is no GPG security key found and asking you if you want to install anyway; ignore them and choose "yes to all". The problem is not with the programs, but with the fact that you don't have the public keys that the programs were signed with on your keyring. This can be fixed later, if you want.
As for Mozilla, even with your sources updated, you may only find a version older than the one you want. You may also want Firebird or Thunderbird. These specific programs can be "installed" from their mozilla.org download pages, because they are usually not installed so much as extracted and run from the locations to which they were extracted. Be aware that there seems to be a conflict between Firebird/Thunderbird and Mozilla in that if Firebird/Thunderbird are installed, Mozilla won't run (either if one of the other programs is already running, or at all-- I can't tell, but I've had this problem under 4 separate distributions, so it's real). Otherwise using them is not a big deal, except for needing to create your own launchers.