If I search my deck for an Item and a Fighting Pokemon with Korrina, can I just take one if I don't have the other?
Yes. Do as much as you can. However, remember that you can fail any deck search for a specific card or type of card voluntarily.
Source: Furious Fists FAQ (2014-08-28), TPCi Rules Team (2014-08-28)
If I attach a Basic Energy to Eevee and trigger its "Energy Evolution" ability but choose an evolution belonging to Team Plasma, would I take the damage from Frozen City stadium or not?
The mandatory effect of the Frozen City stadium comes before the optional effect of Energy Evolution. If the Eevee is not a Team Plasma Pokemon before choosing whether to use Energy Evolution, you would take the damage from Frozen City.
Source: TPCi Rules Team (2014-08-28)
If I only have one card to shuffle into my deck, can I use Maintenance on the principle of 'do as much as you can'?
In this case, the card specifically prohibits you from using it unless you have two cards to shuffle into your deck.
Source: Furious Fists FAQ (2014-08-28), TPCi Rules Team (2014-08-28)
If Training Center is in play, do Mega Pokemon EX get the bonus HP?
No. Although they are Evolved Pokemon, they are not Stage 1 or Stage 2 per the text on their cards.
Source: Furious Fists FAQ (2014-08-28), TPCi Rules Team (2014-08-28)
Can I use Evosoda on a Caterpie the first turn of the game or the turn that Caterpie is played since "Adaptive Evolution" says it can evolve in those cases?
No, you can't use Evosoda to evolve in either of those conditions because the text on Evosoda specifically prevents it.
Source: TPCi Rules Team (2014-05-29)
When playing Lysandre, who chooses which of the opponent's Benched Pokemon get brought forward?
The card says "Switch one of your opponent's Benched Pokemon..." so the person playing Lysandre gets to choose.
Source: Flashfire FAQ (2014-05-08), TPCi Rules Team (2014-05-08)
Can you use Iris when your opponent has taken no prizes?
Yes, because it's possible for your opponent to take a prize during your turn if one of your Pokemon gets KO'd.
Source: TPCi Rules Team (2014-01-02), TPCi Rules Team (2014-05-08)
Does the effect of Iris occur when you play it or when you attack?
Iris' effect looks at the game state at the time of the attack, not at the time when you play it.
Source: TPCi Rules Team (2014-01-02), TPCi Rules Team (2014-05-08)
Can I use Evosoda to search for a Mega Pokemon?
As long as it evolves from the Pokemon you choose, yes.
Source: XY FAQ (2014-02-06), TPCi Rules Team (2014-02-06)