Can I use Weedle's "Speed Evolution" attack on my first turn of the game or the turn it comes into play?
Yes. An attack can override the normal rules for evolution.
Source: Unleashed FAQ (2010-05-13), PUI Rules Team (2010-05-13)
Can I use Weedle's "Speed Evolution" attack to search my deck for Beedrill and place it on Weedle?
No, you can't choose Beedrill from the deck. Beedrill states it evolves from Kakuna, so it is not a card that evolves from Weedle.
Source: Unleashed FAQ (2010-05-13), PUI Rules Team (2010-05-13)