No, the installer file is set to not require administrator rights. You should be able to run the installer at your workplace.
The installer handles all the complications related to which version of Excel you are using and allows for easier upgrades. This is a COM add-in, not a VBA add-in.
Yes, the free trial contains all features of the add-in. There are no restrictions on the free trial other than the length of its use.
You can purchase a license key to continue using the add-in or you can uninstall the add-in from Windows or by disabling the add-in from within Excel.