Moving over to the Game Room.
If the move isn’t level, will you re-move it?
I’ve played backgammon for money, and prefer to play with those who take games seriously even when there’s no stake. Obviously “party” players will have a wholly different perspective.
This. Any rules are plausible but must be agreed in advance.
My son and I play backgammon and have agreed that the dice must be level and can be anywhere on the board, its enclosure, or its checkers.
In serious backgammon play, the dice must land on the same designated “table” (half) of the board, normally the table at player’s right. Permission may be requested before the roll, to cast them in the opposite table. (This request is frequent in the ending when the designated table may be full of checkers.)
When one die is cocked, all dice are rethrown.
Not only must you hesitate before picking up a legal throw, you must hesitate before picking up cocked dice so that other players can confirm the dice are cocked. The question “do you take it on trust” suggests Lord doesn’t play strangers for money. 
The option “demand/ask for a reroll” shows severe misconception, for reasons left as an exercise.
ETA: I didn’t vote in the poll; and despite the excessively precise and pedantic answers above, I admit the question “the die hits a player while rolling” has never arisen in my games, as far as I can remember!
If it’s a game, and there’s no money riding on the outcome, or any other thing of value, and no your pride is not worth anything, then you re-roll any time it doesn’t look like a clean roll. You don’t get all worked up over it however it goes because it’s just a game, and it will never be over with if you spend all your time arguing over whether or not to re-roll.
If it’s a wager, all rolls count and you never re-roll unless everyone is in unanimous agreement that a re-roll is needed. Otherwise, even if the die is precariously balanced on a corner with three faces sharing the topmost position, you have to just sit and wait until it falls flat, even if that doesn’t happen until the next earthquake.
This why no one ever wants to play games with you: you’re always in such a hurry!
Dice that don’t land level are re-rolled. There is no point in re-rolling the others that were valid.
Dice that fall off the table are played as they lie unless the roller calls for a re-roll before a result is clear. Then only that die is re-rolled. I’m actually surprised anyone has ever played these scenarios differently.
Something else: when they get lost. Or when there’s no way to look at the die without moving it. Or if the dice weren’t actually rolled.* Otherwise the randomness is not affected.
*I guess that could be termed as either opponent or player not being satisfied that it’s random, but it’s really about whether the dice were dropped rather than rolled.
EDIT: Missed the one about there not being a number on top. Though I don’t recall it ever happening. Dice don’t tend to land crooked.
I picked the two most common choices (floor and no clear top face), plus “other”. The player rolling the dice can call “that doesn’t count” due to any sort of interference, so long as the call is made before the dice come to rest.
As to re-rolling only a problematic die versus rerolling all dice, I have no strong preference one way or the other, so long as all rolls are made simultaneously, and the standard is agreed to before rolls. If they’re not rolled simultaneously, then you only reroll the problematic die. Back when I was young, it was the established principle among some of my friends that you rolled 3d6 by rolling one at a time, and any mis-roll required re-rolling all three, which meant in practice that if the first two dice were anything other than a pair of 6s, the roller would always “accidentally” drop the third die on the floor.
It really depends on the board game. In Escape from the Curse of the Temple board game, if a die rolls off the table, it’s lost from play.
Go 500 has a special rule if the dice land stacked on top of each other.
I bought a shadow box at Michael’s and lined it with felt and use that as the dice tray. It’s pretty smooth.
That said, if there aren’t any rules about it, any die that flies off the table or lands cockeyed gets re-rolled. The other die/dice are static.
The most important thing that strikes me is if the die is unreadable unless picked up, e.g. knocked under the couch or something. Those ones are so easy to accidentally or deliberately misread. After that, if the dice isn’t level, obviously you have to reroll it. After that, I don’t care–we can reroll if anyone else wants to, but otherwise it’s fine.
Why are you people playing games with people you can’t trust?
That’s what I’m thinking. I understand septimus’s idea of playing for money, and sure, there need to be clear rules there. But I play games to blow off steam, to relax. If I think there’s a decent chance someone’s going to cheat, it’s not going to be relaxing to play with them, so I won’t. Also I try not to care enough about the game’s outcome: if someone cheats, they’re an asshole, but the game itself isn’t that important. So in any circumstances, if I think I need to reroll, I’ll check with the other players there (“It’s on the floor, mind if I reroll?”), and if someone decides to reroll, I’ll go along with it.
I chose two, and they happened to be the two most popular. So, no major arguments in my world when playing a game with dice.
YAHTZEE!!!
Even with no money and full trust, ambiguous rules can lead to problems. Suppose I roll a slightly cocked good number, and tell my son the dice are cocked and I must roll again, but call his dice cocked only when they’re bad. My son would realize I’m helping him win and he doesn’t like that.
Of course, arguing whether dice are cocked or not might add a dimension of personality to the game that some players would enjoy. 
In my house, re-roll ONLY if dice do not land in a position where there is a number clearly on top, or if some dice actually get lost during a roll (fall through a grate or under an immobile piece of furniture/appliance). No need for rules about interference or rolling off the table, or that it has to be perfectly level!
Our rule for that: if you can place another dice of the same size on the ‘top’ face, and it doesn’t slide off, then the roll stands.
We have another rule that says if you roll the dice off the table three times in one game, you automatically lose. This rule was invented for one guy in particular who loves to give his dice a lot if energy.
So, rolls of a d4 are always invalid, then?
D4s are never cocked, so the rule doesn’t apply.
Well, rarely. I guess I’ve never had to deal withthat .
We usually use a box so that the dice are contained. Any dice that pop out of the box are re-rolled.
Also, if a player drops the dice in the box so that they don’t tumble, they are re-rolled.