Can I edit my Final Cut Pro project from an external drive?

Yes, just drag and drop the library.

After you’ve copied the library change the cache location to the external SSD.

For an excellent introduction to FCP start with Steve Martin’s 60 minute intro https://www.youtube.com/watch?v=HqUP7Zgeuck. For a more in depth introduction use the IzzyVideo tutorials. The version is a bit dated but the principles are the same. 

Yes, you can, but why is your library so large if the media isn’t in it? Delete generated library files from the Files menu. Switch off background rendering.

How much free size is on the stem drive? How big is it?

No. It only deletes generated files, render, optimized, proxy. You can rebuild them if you really need them, most users don’t. Make sure the cache is external, on the media drive. Select the library and properties will tell you where the media is.

Delete the generated files to get rid of the optimized media. All your media is stored inside the library bundle. Copy the library to your external drive. Check it. If it’s okay delete the library on your system drive. Because the media will now be on the external, make sure it’s a fast SSD. In library settings set the cache to be on the same external drive as the library. It’s useful to have the cache outside the library. If there’s any problem or the cache starts to get large it can simply be dumped, and the application will rebuild it.

No. That’s garbage.

That says your settings is to have the media in the library. Look at the bottom of the properties pane to see where the media actually is.

Did you delete generated files?

Please post a screenshot what you’re seeing there.