As @Alconja states, I think it will be
Found at the unlucky square
Not sure, unlucky number is 13, but that is Weedle
From your pocket I can scare
Pokemon are sometimes called pocket monsters and monsters do scare
With precious metals, my premiere
Some pokemon connected to metal?
A pet that father and daughter share
In pokemon world adults and kids alike train them
Within a ball I make my lair
They are stored in pokeball
My wings are furled, a double pair
So we are looking for a winged pokemon. Double pair usually means insect type.
So my guess would be
Beedrill. It has double pair of wings and the drill fits the precious metals somehow.
EDIT: second set of hints has been revealed
A tree-named teacher does declare
All of the pokemon professors are named like trees
Of your sex, they're unaware
At the start of the game they ask you: "Are you a boy or a girl?"
A catalogue they have to spare
They give you choice of 3 starting pokemon which differ each generation
So that leads me to:
Assuming one of the 3 starter pokemon from first generation, only one of them has wings which is Charizard. It's number is 9 which is considered unlucky in Japanese culture (where pokemon come from)
Only thing I'm not getting is the line "
With precious metals, my premiere"