1. Bro for Life: The first Pokémon you obtain after your starter, or your starter itself, will be your "Bro for Life". (Think Pikachu, or Iris' Axew.)
You may wait a couple routes before catching something, or you can even import an egg to be your BFL, but you must obtain it before the first Gym and cannot catch anything before you get it.
Once you get your BFL, your starter can no longer be used against any trainers, unless your starter is your BFL. HOWEVER, you may use your starter to train up your BFL against wild Pokémon after getting it.
Your Bro for Life cannot evolve, and must travel with you at all times.
2. You can only catch one Pokémon per area (but it does not have to be your first encounter). It is recommended (not mandatory, but recommended) that you have 10 (14 upon reaching Kanto if playing a Johto game) or fewer Pokémon at all times.
It is up to you as to how you want to handle any Safari Zone, as well as places that have multiple areas (i.e. Pinwheel Forest in BW, Reversal Mountain in BW2, Route 8 in XY).
3. Evolution:
Each possible evolution has a 5/12 chance of happening.
Whether or not an evolution happens is determined by a random number generator. If you catch a Vulpix and the evolution fails to go through, too bad. You have a Vulpix forever.
If you have something that can evolve twice (like Charmander), and the first evolution fails, you may try it one more time (but you cannot attempt to go for a Charizard). That rule is in place to keep the 5/12 number intact.
Pokémon immune to this rule: Magikarp, baby Pokémon, early game Bug-type lines (i.e. Caterpie, Wurmple, Scatterbug). However, for baby Pokémon like Budew and Azurill, after they evolve, then you RNG for their chance of evolution.
4. Departure:
For each new numbered route you enter after the first Gym, there is a 3/16 chance that one of your team members will leave, again determined by random number generator.
If that happens, you must select a member in your current team to release or permanently box, not counting your Bro for Life.
5. No healing items in battle. This is to go along with the anime, which doesn't use them at all. (Although the OP used this rule because he "was a glutton for punishment".)
6. All Pokémon can only learn one TM move for the entirety of the game. The exception is that if the Pokémon has evolved twice while owned by you, you may teach it 3 TM moves.
HM moves do not count towards this limit, nor do TM moves learned by level.
As for Move Tutor moves in Platinum or Black 2/White 2, it may learn one of those as well.
So, for example, say you're playing Black 2/White 2 and are using a Frillish. You could give it Blizzard (TM), Giga Drain (move tutor), Surf (HM) and Ominous Wind.
Additionally, if you import in an egg to be your Bro for Life, you may give it one egg move.
7. Consecutive moves:
You must use three different moves on each Pokémon's first three turns in a battle. If you are early in the game and your first Pokémon only has two moves, it is excluded from this limit.
You can never use a move three times in a row.
(So you'd have to do something like ABCCBCCBCCB and so on.)
8. Gym Battles:
You must begin a Gym battle with the same number of Pokémon as the Gym Leader. You may deposit your Bro for Life before a Gym battle if you do not wish to use it in said battle.
It is up to you how you want to deal with the Elite Four (since the Champion always has six Pokémon).
Optional:
9. No held items, except for those that affect experience/EVs.
To make that more specific, the following items are allowed whether you use this rule or not: Amulet Coin, Macho Brace, Power items (Bracer, Anklet, etc), Exp. Share, Lucky Egg.
Since Gym Leaders/Elite Four in most games give their aces berries, you may give one of your Pokémon a Berry when facing them.
However, as a mercy rule for XY players, the Exp. Share and Pokémon-Amie will be allowed. The former to help during training (and just because), and the latter because Amie emulates what the anime does between Pokémon and trainer bonds including dodging attacks and hanging on.
10. Tweaking rules.
If you ever feel like one of the rule(s) are a bit too much, you may add some sort of alteration. This could be to either simulate the Anime further, or if you ever have problems during a game.
