Final Cut Pro

It all looks like it did install.

I am thinking that you may have a problem with Spotlight.

Try resetting the Spotlight index, and then search for Final Cut.

To reset Spotlight, open Terminal and paste the following:

sudo mdutil -E /

type your password when requested (there is no feedback as you type, not even *** or •••) and press enter.

Give it a while and search.

Just to make sure: FCP is a rather large application, about 3 GB. Did you allow enough time to complete the download?. What do you see in the App Store window? No progress indicator? It should show a circular icon gradually filling around the contour.

Between the previous step and this step was nothing happening in the Applications folder? There was no progress bar or and indication that anything was happening? Immediately after you clicked Install, this window popped up? If that's the case you need to take the system to an Apple Store as the Apple installer software is not accessing the system resources it needs.

You mean it doesn’t download from the App Store? Is there any progress displayed in the Applications folder?

If nothing happens after you click Install I would think you need to take the system to an Apple Store or service technician.

FCP trial version

Final Cut Pro - Free Trial - Apple 

That one or a different link?

Step by step tell us exactly what happens. Post screenshots.

And the rest of it?

You click Continue, and then...