You should not just delete the file to uninstall. I have never used Puppy, but there will be some utility to remove installed programs.
In regards to unpacking, first and foremost, DO NOT EVER run your machine as root. Root should only be temporarily invoked to install software. It is particularly dangerous to run as root in a graphical enviornment, as anything you interact with while browsing can have unlimited effect on your system, ala windows.
You should run as a normal user, without root providledges. Then to extract you can untar/unzip the files anywhere in your home directory, it makes no difference to the system where you do it. You can install the files via root in the console.