I would suggest to check the installation directory for a readme or an uninstall script but in general: do what you did to install it in reverse order and delete everything you added as a setting or as a file on your system related to that installation.
Installations done with a tar file are mosttimes uninstalled with the rm command. You go to the directory where you went for this: "Unpack the downloaded Tar file, android-studio-ide--linux.zip, into an appropriate location for your applications" and rm the files and directories created here.
If you did "You may want to add android-studio/bin/ to your PATH environmental" too you will need to undo this alteration by deleting android-studio/bin/ from the file you added this PATH too.
Short answer
You need to run it via the studio.sh, every time, as that is the startup script.
Long answer
It will not truly 'install'. It is kind of an annoyance for first time users, but it is truly a stand-alone package, it will drop the SDK in your /home and you need the android-studio and jdk1.8.0_45 folders at a fixed location.
First, you need to set the JAVA_HOME path in your /etc/environment or ~/.bashrc configuration to the jdk1.8.0_45 folder before it will run.
After setting your JAVA_HOME, run studio.sh again and it will boot the IDE. Then at the menu, hit the Configure button then hit Create Desktop Entry. You should now have a start menu icon, and desktop icon which you can easily use.
Next time you boot and create a project it should download the SDK required for building.
Best Answer
To create a desktop file do this:
Open Gedit.
Paste the following into the file, editing the relevant parts:
Now, save this file as somename.desktop, to your desktop.
Next, you need to make this file executable by typing the command:
Now, double-clicking the application should launch your application.