If the Broken Time-Space stadium is in play, can I use Rare Candy on a Basic Pokemon that just came into play this turn?
No, the wording of Rare Candy specifically says "you can't use this card during your first turn or on a Basic Pokémon that was put into play this turn". Broken Time-Space does not allow you to override that.
The wording of Rare Candy is now as follows, "Choose 1 of your Basic Pokémon in play. If you have a Stage 2 card in your hand that evolves from that Pokémon, put that card on the Basic Pokémon. (This counts as evolving that Pokémon.) You can't use this card during your first turn or on a Basic Pokémon that was put into play this turn." So now it can't be used on your first turn, and it can't be used on a Pokémon played this turn.
Source:
TPCi Announcement (2011-04-11), PUI Rules Team (2011-05-05)
The wording of Great Ball is now as follows, "Look at the top 7 cards of your deck. You may reveal a Pokémon you find there and put it into your hand. Shuffle the other cards back into your deck." This card works very differently now; it can no longer search the entire deck for a Pokémon and put it directly onto Bench.
Source:
TPCi Announcement (2011-04-11), PUI Rules Team (2011-05-05)
The wording of Defender is now as follows, "Attach Defender to 1 of your Pokémon. Discard this card at the end of your opponent's next turn. Any damage done to the Pokémon Defender is attached to by attacks is reduced by 20 (after applying Weakness and Resistance)." Defender now protects against all attacks, even ones not made by the opponent.
Source:
TPCi Announcement (2011-04-11), PUI Rules Team (2011-05-05)
Can I use Rare Candy on a Pokémon's first turn in play? For example, can I place down a Wurmple and then use Rare Candy on the same turn to evolve it into Dustox or Beautifly?
No, this is no longer permissible [per BW rules changes].
Source:
PUI Rules Team (2003-09-11), PUI Rules Team (2004-07-22), PUI Rules Team (2011-04-11), PUI Rules Team (2011-05-05)