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

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

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