Questions about the way that OS X handles files.

You can't delete something that is in use. You need to quite the copy that is running (which presumably is living on your desktop) and use the copy in your Applications folder.

 

If you drag an App to your desktop in Lion, you usually get an Alias, rather than a copy of the App. If the icon on the desktop has a little arrow in the lower left corner, that means it's an Alias, a file that points to the App.
To move a COPY of the App to the Desktop, hold down the Option Key while dragging.
To move the App itself out of its folder and onto the desktop, hold down the command key while dragging.

Logging out, and back in (don't preserve App states) usually fixes an ornery trash-can.

 

Do you WANT to lose that application? Not clear...

If not, suggest you put it back in the Applications folder. (It might ask you for your admin password if, as recommended, you do your everyday computing in a non-admin account.)

If you do want to be rid of the app, either command-tab to bring it to the foreground and quit it, or use the Apple menu, select Force Quit, and quit it there, or use the Activity Monitor to find it and force-quit it there (but be careful you know what you're force-quitting in that case).

Then you can drag it to the trash. This uninstalls it. Easy, huh? ...It does not need to make a pit-stop on the desktop; you can drag directly from the Applications folder to the trash. Again, if you have your machine configured so your everyday work is performed in a non-admin account (recommended), you'll be asked for confirmation.

As to your other question, yes you can have two identical icons representing two different applications, or two different versions of an application. Usually, however, when you install a new version of an app (generally by dragging it to the Applications folder), you'll be asked if you want to overwrite the old version. Most often the answer to this will be yes.

The one thing to beware of is whether the multiple icons you're talking about represent different applications or aliases (pointers) to the same application. You can command-I each of them to see more info about them.

Welcome to OS X... it'll all start to make sense as you use it. It's a fabulous OS.