For VR purposes, Pickup will pick up just about anything except evolutionary stones (and maybe very rare items like Light Balls). I would estimate a rate of one item a day. Natural items like berries, miracle seeds, hard stones, and honey are more likely in natural areas, while in urban areas you're more likely to find manmade items like Potions, Pokeballs, and so on.

As for your second question, Pokemon you hatch are automatically registered to you! As Pokemon with Trainers are generally excited about the Trainer-Pokemon relationship, Thace can probably expect his newly hatched friends to dig Pokeballs out of his bag or even get jealous of other Pokemon's Pokeballs.

Remember that the maximum number of Pokemon a person can have with them at any time is six. More than that, and you'll have to send one or more to the PC, which DEFINITELY requires a Pokeball to do!