Say you are using a Golduck (from the Fossil set) and you use his Psybeam. Would this be a Psychic or Water attack?
A Water attack as Psyduck is a Water Pokémon. [Ed. Note: This answer can be generalized further: When a Pokémon attacks, that attack is considered to be an attack of that Pokémon's type; it's not dependent on the energy requirement.]
If I use Pokémon Trader or Energy Search and there aren't any Pokémon or Energy in my deck, do I have to show my deck to my opponent? If there isn't any in my deck, do I get to put the Pokémon Trader or Energy Search back in my hand either?
You don't have to show your deck and you don't put the trainer back in your hand [Ed. note: It goes into the discard pile].
This is probably an incredibly silly question, but I can't find the answer to it anywhere. If an opponent knocks out one of my Pokémon, and they forget to draw a prize, are they entitled to it if they remember later?
Technically yes. It is the responsibility of BOTH players to make sure everything is done correctly. If you spot it and he forgets you can BOTH get a warning in a tournament if you don't catch it.
What happens if I use Professor [Juniper] and have less than 7 cards left in my deck? Do I lose immediately, or do I draw out the remainder of my deck and lose at the beginning of my next turn (if I don't do anything to replenish the deck like Recycle, Mr.Fuji,
You only lose when you don't have any cards to draw on; the card draw you must do on the start of your turn. So you would lose on your next turn if you didn't somehow get some cards back into your [draw] deck.