I have Firefox plugin SQLite Manager installed on my computer and have SQLite database.
Is it possible to open SQLite database by double click on it with SQLite Manager within the Firefox add-on?
No. For that you would need a command line handler and the SQLite Manager extension doesn't implement one. You can ask the developers to implement one so that their extension can be triggered from the command line - or you can do it yourself of course. The code example in the documentation is unfortunately outdated and needs to be adapted to the XPCOM changes in Gecko 2.0.