In Pearl, I caught a Shinx in the route above Sandgem town. I named him Shadow because Luxray's hair reminds me of Shadow's quills, lol.
Either after capturing my first Pokemon or after finishing some battles, not sure, I also got Pokerus that early in Pearl.
As for HGSS, my first capture was a Pidgey. I named him Peanuts, after my original Pidgey in Gold.