Yes, you can have golden versions of the cards you get from playing the Curse of Naxxramas adventure mode. This was confirmed by Community Manager Zeriyah over at the Battle.Net forums. However, you can only craft the golden version of the card if you have already acquired the normal version
Additionally, all cards will be available in Arena Mode even if you don't open up a wing and beat the bosses. That means that once Arachnid Quarter opens, all cards that can be acquired from it will show up in Arena mode.
<source>
.
