Is it possible to "install" a module just by pasting the file of the module and just paste it into the DesktopModules folder and have the modules working on the website?
If not, is there a way where I can export the whole module from one system and install it onto another system?
Have you tried creating a module package for this module? (when you click the edit pen for the extension, you should find a "Create package" button on the top right).
Take care to include the assembly/assemblies needed, and also the Sql Server scripts to install the module correctly (often found in the module folder under Providers/DataProvider/SqlDataProvider/*.SqlDataProvider). There might be other files in the module folder (images, css, etc), be sure to include them in the package, then try to create it. You will find the ZIP file under /install/Modules then, move it away from there somewhere on your local file system (otherwise the installer might try to install it with the next update). Then try to update the extension in a testing system, check if it works and you're done. If there are errors it might be that you are missing some files that have to be included. Check the error messages, and restart.