If I have Klefki with "Mischievous Lock" Ability active and switch or retreat it to an Iron Valiant ex with "Tachyon Bits" Ability, can I use Tachyon Bits?
Yes, when Iron Valiant ex switches in Klefki is no longer there, so Mischievous Lock does not prevent the use of Tachyon Bits.
Source: TPCi Rules Team (2023-11-09)
What happens if Klefki with "Mischievous Lock" and Empoleon V with "Emperor's Eyes" Ability somehow both become active at the same time, like at the start of the game?
If Abilities activate at the same time, the Abilities of the player whose turn it is activate first.
Source: TPCi Rules Team (2023-04-06)
If Empoleon V with "Emperor's Eyes" Ability is in the Active Spot and my opponent brings up a Klefki with "Mischievous Lock" into their Active Spot, whose abilities are turned off?
Since Emperor's Eyes was in effect before Klefki became active, it takes precedence and Mischievous Lock does not activate as long as Emperor's Eyes is in effect. The converse is also true, if Klefki were active first and Empoleon V is brought up, Emperor's Eyes would not activate.
Source: TPCi Rules Team (2023-04-06)