MacOS – Why do applications I install only sometimes appear in the Applications directory

What I do in these cases (which I seem to run into fairly often lately) is create a new folder in Applications, and then drag everything from the disk image into that folder. It's one extra level, but it works just fine for almost every application.

This icon is shown on files that have been downloaded from the Internet and not used yet.

The first time you open the application, you will see a dialog warning you that you downloaded the application from the Internet and asking if you are sure you want to run it.
Say yes and run the app, and the icon should soon be changed back to what it should be (you may need to log out and back in).

To answer your question in general:

To change an app's icon, you first need to copy the new icon you want to use to the clipboard.

1. Open the image file you want in Preview (found in your Applications folder),
   select all (⌘+A), and copy (⌘+C).

2. Find the app whose icon you want to change. Select it in Finder and choose File->Get Info (⌘+I).
   

3. Select the small preview icon in the top left of the Get Info window, and paste (⌘+V).