Hey @Jana_Sokolović_v2,
if you save the search reports you can easily access them via the left hand side.
If you want to create a separate project from it you might want to export the project data via csv and import into a project.
You could also use Asana2Go for this
The only thing is that it would create new tasks basically.
So you might be better off with multi-homing the tasks from the search result to be able to work on them in the separate project also and all changes done will reflect everywhere.
Does that help?