Is there any way to run external program or command after I open project in WebStorm?
I've created new external tool which runs database server, and after I click on it in menu, it runs the server and everything works, but I'd like to start this tool automatically on project start. And it'd be awesome if it could also open web browser to certain url which would be database admin tool.
If you can launch your desired task using Grunt/Gulp/npm task or any other Run/Debug Configuration then yes -- there is Startup Tasks functionality available at Settings/Preferences | Tools | Startup Tasks.
Note that you cannot execute External Tools entry directly or launch any DOS/shell command this way.
The easiest way of using this would be creating Grunt/Gulp task that would do your actual job. There you can use exec() method to execute custom command.