Does Gengar & Mimikyu GX's "Horror House GX" attack prevent me from using Viridian Forest?
No, it does not. Discarding a card from your hand is not the same as playing a card from your hand; the card goes to the discard pile, not "in play".
Source: TPCi Rules Team (2019-08-22)
Can I use Viridian Forest if I have no cards left in my deck, and just discard a card from my hand?
Yes, you can. The first sentence of Viridian Forest 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)