Whenever any cards move from a Zone anyone can see (such as the Discard Pile) to one not anyone can see (such as your Hand), that card must be revealed to all players.
Can I use Professor Elm's Training Method if there are zero cards left in my deck?
No, you cannot search through your deck if there's nothing there to search. If there are cards in the deck you can fail to find a Pokemon because you don't know what may be in a deck, but since you do know whether there are ANY cards or not you cannot search through an empty deck.
Can I use Mew's "Re-creation" attack to choose a Lugia LEGEND card with the attack "Elemental Blast" written on it from my opponent's discard pile, and use that attack?
No, you can't use any attacks from a Pokemon LEGEND unless both pieces are in play.
If Primeape's "Top Drop" attack reveals a LEGEND Pokemon card with no HP printed on it, how much damage does it do (if any)?
No damage is done regardless of which LEGEND half is revealed by Top Drop, because even the half with HP written on it is not considered "a Pokemon" while in the Discard Pile.
Source:
PUI Rules Team (2010-02-04), PUI Rules Team (2010-02-18)
What if during setup the only Pokemon cards in my hand are Pokemon LEGEND halves?
Since you have no Basic Pokemon to place Active you will let your opponent look at your hand, then shuffle and redraw your hand as described in the rulebook.
Source:
HeartGold & SoulSilver FAQ (2010-02-11), PUI Rules Team (2010-02-11)