Of course they would. Icahn couldn't actually run Yahoo. So if they were negotiating with him they'd be negotiating with someone who didn't have the option of remaining independent.
Absolutely. I love their statement that they will not negotiate with Yahoo's current board. I interpret that to mean that Steve Ballmer does not like to negotiate with people who say "no" to him.
Icahn's whole point is to make a couple of billions from this transaction, he has no interest whatsoever in Yahoo or the shareholders.
Why on earth would a company want to sell just the most valuable part of its business instead of selling it all?