Why is VLC verified every time I open a file that is supposed to open with it? I have downloaded VLC for Mac, and I installed it. Now every time I open a video that is supposed to be opened by VLC, it starts verifying again, like it is the first time and it will install it. I also don't see a shortcut in the Applications folder. What is the way around this?
If your Applications folder doesn't contain VLC.app, you probably didn't install it properly. The verifying process (trying to start VLC) indicates that a dmg containing the app has to be opened.

Since VLC is a self-containing app you don't have to initiate an installation. Instead open the downloaded VLC.dmg, drag and drop the VLC.app inside the mounted dmg to the folder /Applications. Then unmount the dmg and start VLC.