The artifact card you find is exiled face up. All players can see what it is.

If Hoarding Dragon is put into a graveyard before its first ability has resolved, its second ability will trigger and do nothing. Then its first ability will resolve. If you choose to find an artifact card from your library, it will be exiled for the rest of the game.

Hoarding Dragon’s two abilities are linked. The second ability refers only to the card exiled by that Hoarding Dragon’s first ability. In other words, each Dragon has its own hoard.

If Hoarding Dragon dies before its first ability resolves, its second ability will trigger and do nothing. Then its first ability will resolve. If you choose to exile an artifact card from your library, it will be exiled indefinitely.