How to achieve the best outcome by a single statement in this game?
A caveat, this is not really an answer, more of an extended comment--that's why I've put it as community wiki. This is a cool game, one of commitment.

I'll try to formalize the problem. This is a first pass, so people should edit as they see fit.

It's probably easiest to solve the case where, $T$, the number of rounds before a forced execution is $1$. Let each player get $1$ if they shoot someone else and $-1$ if they are themselves shot.

There are two stages of the game, the commitment stage and the shooting stage. The sequential stage is (fittingly) sequential move and the commitment stage is not really a game persay, since the players are automata (their strategies are predetermined). I'll return to this shortly.

An easy result: there is an equilibrium in which rules 2 and 3 are rendered superfluouous since each player can commit to every possible contingency. 

There is a source of exogenous randomness, a random state of the world $\Theta = \Theta_{1} \times \Theta_{2}$, $i = 1,2$, where $\Theta_{i}$ is the set of permutations of $\left\{A,B,C\right\}$. These are the possible orders in the shooting portion.

The set of actions for player $A$ is $\mathcal{A} = \mathcal{A}_{1} \times \mathcal{A}_{2}$ where $\mathcal{A}_{i} = \left\{B,C,\emptyset\right\}$ for $i = 1,2$ (corresponding to "shoot $B$," "shoot $C$," and "shoot nobody," respectively). The sets of actions for the other two players, $\mathcal{B}$ and $\mathcal{C}$, are defined analogously.

A (mixed) commitment strategy of $A$ is a mapping $$\sigma_{A}: \Theta \times \Sigma_{B} \times \Sigma_{C} \to \Delta\left(\mathcal{A}\right)$$
where $\Sigma_{i}$, $i = A, B, C$ is the set of all (feasible) mixed strategies. An example of an infeasible strategy for player $A$, say, is one that does not have $\emptyset$ played with probability $1$ when the state and mixed strategies of the other players are such that $A$ has been shot for certain.

Note that I am using the fact that the shooting portion of the game can be viewed as a instantaneous (random) outcome of the commitment portion, which is the real game.

I'm not quite sure about the commitment strategies for $B$ and $C$, due to the sequential move nature of the commitment stage. I suspect that that does not matter, and that they are defined analogously. I'm not completely sold on this and open to being convinced otherwise. 

Each player's payoff is defined in the obvious way.