This is my understanding of the Dragonballs. If I am in error, 
someone please correct me.

1. Each planet can have its own set of Dragonballs.

Only Namek was supposed to have them, but since Kami, and later Dende was
from Namek, he brought the idea over.

2. Each planet's Dragonballs are supposed to remain on the planet 
they belong to, and can only be used on that one planet.

Not true.  The Namek balls were used on earth during the Freeza Saga.
There just has to be a kami-type figure to power them.

3. In order for there to be Dragonballs, there must be a kami for the 
planet.

This is correct, so far as I know.

4. The kami must create the Dragonballs.

Again, correct.

When Piccolo was killed during the battle with Raditz, and Kami-sama 
died with him, the Dragonballs were lost, since they were the set 
belonging to that Kami-sama. Thus, the Dragonballs were lost for a 
long time, until Dende created his own set. 

Nope.  The Dragonballs became active again as soon as Piccolo was wished
back to life.  Dende later upgraded them so that they grant two wishes,
however.

I think that's how they work. Someone correct me if I made a mistake.

Sure thing.  ^_^