Just a few house rules I use;
Fudge: Sure alot of you have played with a fate/faith points system at one time, as to help out a character that's knocking at deaths door. For those who do not know about or never heard of them; It's a way to save a misfortune befalling a character, be it a failed save versus a death spell or being on the nasty end of a powerful/killing attack roll, ect. I'm a GM who doesn't like to fudge dice rolls AT ALL. Thus, it's why I used this system but I often found myself having too think of something off the top of my head to tell what happened/intervened to save that characters life, "A angel is watching over you., The fates deemed it isnt your time.," or some other hack 'filler' as to pull it all together. I now use the same idea of fate/faith points but cut out the B.S. What I do now is, once per level, I have the characters write the word fudge at the top of their sheets and what that does is allow that player to reroll any one role they made or any one role I (as the GM) have made versus their character. After they use it, they erase their "fudge". Maybe it's because I have been playing for many years now ,as well as many of my players too, that it feels okay to skip on the 'cheese' and go right for the 'fudge'? Let's face it... sometimes them dice are harsh and the players totally understand that too! Just a way to help skip a bit of 'ugliness' and keep the game moving along.
Auto Fails: I also just started using this house rule too. When ever a player tosses out a die and blurts out a number, as to see if he/she hit/succeeded, without stating what that action was first, they automaticaly miss/fail what ever action they had locked up inside of their mind and failed to mention. Of course that's after the fact I have too ask "what did you do?".
Fudge: The concept makes sense to me, though it's something we've always done in a more "behind the scenes" fashion. In my group, we get invested in our characters, both and players and DMs. As a result, we try not to kill PCs in meaningless ways unless it's as a result of extreme stupidity on the part of the player. To that end, the DM sometimes fudges dice in the players' favor.
The players are aware of this, and it doesn't detract from our gaming fun. The trick is to make them feel like they're in a huge amount of danger even when you're trying not to kill them.
Auto Fail: This particular scenario doesn't come up in our groups (different player tendencies, I guess), but I don't blame you for it if it's a regular problem. When I DM, I do hold a "don't let me catch you cheating" policy. I occasionally catch players fudging their die rolls. In such a scenario, they "auto fail" their check.
Auto Fails: I also just started using this house rule too. When ever a player tosses out a die and blurts out a number, as to see if he/she hit/succeeded, without stating what that action was first, they automaticaly miss/fail what ever action they had locked up inside of their mind and failed to mention. Of course that's after the fact I have too ask "what did you do?".
We use action points as seen in Eberron to a greater extent, in that I allow them to burn action points to perform outlandish actions in combat (which really doesn't amount to anything but complex skill checks with lowered DCs for cinematic effect, but the players don't have to know that).
As for player blurting out numbers, I just started last night with telling that they have to declare an action before rolling the die, and I just made him reroll it. It's not been a huge problem, but depending on the circumstances, it can be important, especially when the chance of failure can affect storytelling.