WHoa. A lot of replies! Awesome! I feel I should clarify my original points, rather than go through people's replies point by point - I'd love to, but I'd be here all day!
1. Average Healing Rolls between Encounters
This would be done purely to speed up short rests. I find 4e can get a bit "draggy" at times, and this is one of those things to speed up play.
2. Average Damage Rolls for Secondary Attacks
Again, done to speed up play. I don't think there are too many cases in my group that need to worry about this, but I find it sort of silly that a controller has to use the same damage for all of his area attacks, but a character that makes iterative attacks rolls new damage for each attack. Since my brother made a hypothetical ranger that can get eight attacks in one round, I figured this is a house rule I should declare right now... rather than seeing any single player take up a huge amount of time rolling damage for attacks.
3. Monster Knowledge Checks
Here's the big one. Every fight, I have a character that does a monster knowledge check. When I started my 4e campaign, I said "everything will be core by the rules", and I've been sticking to that, so when he makes his monster check, I have to give the player info on the monster's attacks, role, name, resistances... it REALLY bugs me, because I feel it takes some of the mystery out of the game.
I'm perfectly happy giving clues with monster knowledge checks. And I don't mind giving away full information for relatively common monsters. I really just want to have more power in my hands for deciding what information PCs get in a certain situation.
4. Ignoring Dumb Rules
Okay. I have no problem with the tripping of oozes, or with Come and Get It possibly moving a monster into bad terrain. But I'll give you two examples of why this is coming into play.
First off, remember that I said the game will always follow the rules when I started the campaign. This means that players can read the rules, as written, and do actions that defy "reality". I realize reality is highly subjective in a fantasy RPG... but there are things that really jerk players out of the game.
Situation one was in my game last month, and it bugged the hell out of me. The rules say that if a character moves two squares, he has a running start on an athletics check. In this situation, the player essentially moved his character back one square and then forward one square to get the two square running advantage - even though "realistically", everyone at the table agreed it made no sense (I can't remember the exact situation, but it was something that would not work in real life... it wasn't just getting room for a "running start").
Situation Two was in a game I was playing a few weeks back. Again, it made no sense thematically, but it followed the letter of the rules. Essentially, we knocked a gnoll prone, and one of us tried to shoot past the gnoll to the archers behind him. Unfortunately, the prone gnoll still granted cover. This was a "wtf?" moment for the group. To be fair, the GM follows the same rules, so it wasn't like it was intentionally unfair or anything, but it was one of those things.
My essential purpose for this house rule is essentially to say "I know I said we'd follow the rules exactly as written, but there are times that doesn't make sense, and I want the freedom to make those rulings without players getting mad at me for changing my approach halfway through the campaign".
5. Flavour for "Stunting"
There's a tendency in 4e (at least among my group - this isn't an edition war!) for players to simply read off the power name, and roll some dice. "I use come and get it" and then tell me the mechanical results. I want players thinking more about their powers, and get them a bit more involved in the thematic side of the game, and I think this house rule will encourage that. One player has already been doing this, and I really like it. Essentially, I'd be willing to let a character with Come and Get it possibly knock a few people prone with a modification to the power on the fly, or the bard using his shout power to break a window over the head of some minions and thus make the area beneath them dangerous terrain.
6. Takebacks on Moves
We have a few players who, when it's their turn, consider every possible movement choice. They will move their character, count ranges, move their character back, and so on. In the game I'm playing in now (which is all the same players, just a different GM), this rule was implemented, and I like it.
Essentially, there's a slight time limit on your turn, and once you move into a square, you're there. Takebacks are only allowed if you moved in a certain way assuming information that your character didn't have (That guy you wanted to target was actually invisible, for example, so your character wouldn't have moved to attack him).
The goal is to speed up play in combat, and it's worked well so far. I can see why a lot of players would hate this one, but I actually think it's one my current group would barely notice.... all it would do would encourage one or two players to take quicker turns, without anyone in the group actually having to say something to those specific players and possibly embarrass or hurt feelings.
7. Backgrounds
It'd be kind of lame to allow players to choose backgrounds for characters they've been playing - in some cases - for over a year. But I like the mechanic of backgrounds, and want to see it in play. So, since I'm moving the campaign to Sigil, I think a bunch of "Faction" backgrounds would work nicely. I'm going to explicitly state this to the players with the other house rules. But I'm toying with how to do it exactly... my current plan is to have each faction have multiple powers to choose from, but to not tell the players what those powers are until they join the faction (so they're not joining for a purely mechanical benefit, but instead for RP reasons).
8. Leader Abilities to Include Self
Someone on the class survivor threads pointed out that powers that do not include self are self-defeating. Essentially, a group buff makes the party stronger. A character that does this as his "schtick" improves the group's overall effectiveness... meaning that if the GM wants to throw balanced encounters at the group, he'll up the XP budget or throw better-constructed encounters at the party.
However, if the leader's powers do not include self, what he's done is upped the danger level to the group. Sure, he's improved the group's ability to respond to that danger, but he has not improved his own ability to respond to that danger. He's essentially nerfed himself, and paid character resources for the privilege.
Right now, my group has one leader - the other leader was retired when I politely asked one of the players to retire one of his two characters (he volunteered for it). With one leader and six PCs, I really don't want the player to just feel like he's the group healbot. This is one I'm going to put into a trial period, for sure, but I want to do something to make leader characters more enjoyable to play, and less like a chore.
(For the record, my group's leader seems to enjoy his character. And I play the leader in our other game, and love it. But we've both seen the ally rule get in our way, and it sucks to blow a daily on making everyone but yourself better).
One final thing - I think some of you might read this and start arguing about whether or not a player should have done something. Or suggest that my players are all bad hacks, or something. My players are all great - I have a great group, and I think their actions have been perfectly acceptable in play. The fault lies with my initial declaration of following rules as written - I should not be angry when a player does so. So I'm not really looking for a "well, that's wrong of your players to do", or anything like that.