How does the game determine what box a caught Pokemon will be sent to

No, you are not using it passively :p.

To use the Pokemon Radar:

• You must be in tall grass. (This includes Flower grass, but unfortunately not the Route 6 grass, caves, or Water).
• You cannot be rollerblading.
• It must be fully charged. (After you use it, you will need to take about 50 steps before you can use it again)

Up to 4 patches of grass will shake. (It technically always chooses 4 spots around you but most patches of grass are smaller than it's range, so it may 'shake' a position that is not tall grass and therefore you will see less than 4.)

If you go into the shaking grass it is very likely that you will fight a pokemon. It is not guaranteed though.

After each battle four more spots will shake. If you choose a spot with the same pokemon as the last one you fought the cycle will continue. This is part of a chain.

Speculation: It looks like some spots will shake with more intensity than others. The less it shakes the less likely the pokemon in it will be the same as the one you just fought if there is any pokemon at all.

You will end the chain if:

• You start rollerblading.
• You run from battle.
• You encounter a pokemon that is not found in one of the shaking spots.
• You fight a different pokemon than the previous one.
• You do not find a pokemon in the shaking grass at all.

Helpful Tips:

• Repel and abilities like it will help prevent random encounters that may break your chain.
• Supposedly choosing the furthest away piece of grass is your best bet to find the same pokemon.
• You can catch any of the pokemon you encounter. It will not break your chain.

http://bulbapedia.bulbagarden.net/wiki/Pok%C3%A9_Radar

In my experience, hunger rates do not change over time, but higher leveled pokepuffs do fill the pokemon's hunger more than lower leveled ones.

However, if you want to go the other way and make them hungry quicker, you can do so by playing minigames with your Pokemon with Pokemon Amie.