In Unlimited, can I use Korrina to get old Trainer cards like Professor Oak, Item Finder, etc.?
Yes. If a card is not a Supporter or a Stadium card, it would be considered an Item card by a new card effect.
Source: TPCi Rules Team (2014-09-04)
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)