Can you make a new movie out of a small clip of a big movie?

Yes, you can make a new movie out of a small clip of a big movie. Select the clip in the timeline. Do an Edit/Copy. Then create a new project and Edit/Paste the clip into the timeline.

Another way is to open a project and click on the project name displayed under the Project Media item at the top of the media side bar. That will display in the media browser all the clips that are in the project. You can select and Edit/Copy the clip from there and Edit/Paste it into a new project.

-- Rich