In contrast to the beta, OS X Final does not offer copy & paste of icons any more. The icons simply don't appear in the information window. It might have to do with my version, since I use 10.0.1 German.
This is a glitch in your system it sounds like, because the icons should be appearing, and you should be able to copy and paste them.
You should possibly try re-installing MacOS X from the original installer and then run the 10.0.1 upgrade again. Delete the Software Update and 10.0.1 items in the /System/Library/Receipts folder before re-installing. It's perfectly OK to install in this manner on top of 10.0.1. It doesn't downgrade much, and when you run the updaters again you'll be back up to the same level.
It sounds stupid to do this on a Mac, but when you've got what sounds like a corruption issue then this is the most thorough way to correct it. And it won't harm your system in any way. If you've upgraded Apache or installed third-party unix-level stuff then you might want to double-check those. But if you're just doing the basics, downloading with Explorer and all that, then there's no important settings you could lose.
Maybe somebody else will back me up on this...? ;-)
Of course problems like this can just go away on their own sometimes too, like if you got more RAM it might suddenly work. Weirder stuff has happened.