Actually, I'm glad you started this because I was going to put something like this up on the site anyway.
Let's see.
Standard stuff:
You can only keep the first Pokemon of an area that you find or are given.
You must nickname all of your Pokemon.
If a Pokemon is fainted in battle, it must be released or boxed and never used again.
Okay, so what else?
Easy Mode:
All places listed count as separate areas.
If you obtain a Pokemon without a battle, you may choose, once per area, to substitute it for the firstly encountered Pokemon on that area.
You may use the daycare.
The battle with Garchomp is exempt from killing your Pokemon.
Normal Mode:
Places listed with italics do not count as separate areas but places listed with asterisks do.
If you obtain a Pokemon without a battle, you may not replace the firstly encountered Pokemon with it.
You may use the daycare to produce up to six eggs total, ever.
The battle with Garchomp is not exempt from killing your Pokemon.
Hell Mode:
Places listed with italics or asterisks do not count as separate areas.
If you obtain a Pokemon without a battle, you may not replace the firstly encountered Pokemon with it.
You may not use the daycare.
You must defeat the Garchomp.
Making the list in a different post because I've already had to rewrite this twice due to derpy computer things and I don't want to make that a third time