All you need to copy is the installer program file you downloaded from Shotcut, or download a fresh copy from the download tab at the top of this page.
You can install multiple versions of Shotcut if you wish on any computer, but using one version to the next with projects may get mixed results. By default Shotcut will not install multiple versions.
Concerning your project MLT files and all of your media files, all of those will copy over just fine.