Is it possible to install SQL Server on an external drive and use it? Also, I am trying to install a software on my clients usb 3.0 flash drive from where they can operate it easily on any laptop or PC - is this possible?
Well, SQL server is installed within an operating system and is dependant on registry entries and possibly other settings within Windows OS installation path. As such, even if you install SQL server on an external drive then attempt to connect that external drive to another machine, it will not work.
The alternative is to use a Linux installation with a Windows OS installed on a virtual machine along with SQL Server.
An other option is to look into a contenerized solution such a Docker.