If my opponent uses Mime Jr's "Mimed Games" attack and they don't have enough cards in the Lost Zone, can I choose Sableye's "Lost Mine" attack and if so what happens?
No, you may not choose that attack, you must select a different attack.
Source: TPCi Rules Team (2024-02-01)
If I have 5 cards in my Lost Zone and my opponent has 5 cards in their Lost Zone, can I use Sableye's "Lost Mine" or Giratina VSTAR's "Star Requiem" attack or Fantina?
No. Each player has a separate Lost Zone above their Prize Cards that they put cards into. You yourself must have 10 cards in your Lost Zone in order to do either of those attacks or to play Fantina.
Source: Lost Origin FAQ; TPCi Rules Team (2022-09-08)