Can I use Giant Hearth if I have no cards left in my deck, and just discard a card from my hand?
Yes, you can. The first sentence of Giant Hearth says to discard a card from your hand. Then if you can do that, you do as much of the rest of the card as you can, which in this case is draw no cards.
Source: TPCi Rules Team (2019-08-01)

Relevant cards

Note: All applicable cards may not be listed below.
SM11_197
SM11_197