If they weren't builtin, and you didn't install them, how did they become part of your system? What instructions are you referring to?
My suspicion is that at some point you did install them, tried to uninstall them, but the 'uninstall' failed in one of two ways. When you install an application, a file is written to /root/.packages. Note the dot (.). It's a hidden file. Using rox, you can left-click the 'eye' on rox's tool bar to 'show hidden files'. Other file-managers may have a menu entry. Builtin files are also noted in a folder named 'builtin_files' within /.packages. The files noting an application's inclusion on your system are plain text files. You can open them in a tex-editor. Their text shows each of the files which the package installed contained and where those files currently reside on your system. I don't have any firefox installed on my system. But a partial listing of the palemoon file --the full listing takes 59 lines-- shows:
/opt/palemoon/libnssdbm3.so
/opt/palemoon/libnssutil3.so
...
/usr/bin/palemoon
/usr/share/applications/palemoon.desktop
/usr/share/icons/mozicon128.png
Puppy package manager writes them when you install an application and reads them when you select "Uninstall" in order to know what files to uninstall. Menu>Setup>Remove Builtin Packages reads the files in the "builtin_files" folder. If you do a remaster, that process should delete entries from .packages after copying them to /.packages/builtin_files.
The module which generates menu entries does not examine that folder. Rather, it only looks for "desktop" files, most of which are in /usr/share/applications.
Problems like you experienced sometimes happen during a remaster, especially if you took the opportunity to edit /root in the 'build folder'. Further complications may arise if you install one version of an application --such as firefox 19-- over another version such as firefox 15 then uninstall one. The files of the second install will over-write the files installed by the first; and the unnstall will remove the only copy of that file.
Regardless of how the problem arose, what it is comes down to is two possibilities: (a) The applications are, in fact, not on your system but Puppy Package Manager sees files in .packages which tell it that they are. This is the easiest problem to solve. Ordinarily, I would tell you to just delete the files in .packages named mozilla-firefox-15.0.1, mozilla-firefox-19.0, and mozilla-firefox-17.0.3esr and any "desktop" file in /usr/share/applications referring to them. But this is firefox and if you have another version of firefox on you system, you want to be careful not to delete references to it.
The second possibility (b) is that the (or some) files from the firefox 15, 19 and 17esr pets are, in fact, on your system, but the files in /.packages referencing them were already deleted. If you do want to have some version of firefox on your system, then I would suggest you download that application but don't install it yet. Rather, run Menu>Filesystem>pfind and enter 'firefox' into the search box. Then file-browse to any file it 'finds' and delete it. Also, delete the hidden /root/.mozilla/firefox and /root/.cache/mozilla/firefox folders. Then install the version of firefox you downloaded. Obviously, if you don't want a working version of firefox, don't bother downloading it.
Or maybe I've just misread your post. The graphic you posted seems to be entries generated by typing firefox into Puppy Package Manager's Search box. What entries do you get if you click "Uninstall" on Puppy Package Manager's Menu?