Cannot empty trash after deleting .dmg
The hard-drive looking thing on the desktop is a mounted disk image; the .dmg file is the file where the data for the image is. 

To get rid of the image, unmount it. There are three four five good ways to do it:


Drag the disk to the trash. The trash will turn into an eject icon while dragging.
Highlight the disk on the desktop and select File > Eject in the Finder menu bar.
Highlight the disk on the desktop and press Command-E.
In any Finder window, press the Eject button next to the disk in the sidebar.
Right-click (or control-click or two-finger-click) on the icon and select Eject. (from JoshHibschman in the comments below)
You can also eject from the commandline using diskutil if you really want.


After you eject the disk

Insofar as the application "file" is concerned, there is only one copy of the application: the one in the /Applications folder. The icon in the dock is only a link (not really a shortcut, as there's no file like a shortcut in the Start menu or the Quick Launch bar, but it's not bad to think of the Dock icon as a "shortcut".) 

And to answer your first question, this is a great place to come with questions. Hope you have a good time on the Mac!