1. There is a general algebraic result which states that the abstractly defined associative $\mathbb{R}$-algebra with generators $X_1,\ldots,X_n$, $Y_1,\ldots,Y_n$ and relations
 $$
X_iX_j+X_jX_i=0, \quad Y_iY_j+Y_jY_i=0,\quad X_iY_j+Y_jX_i=\delta_{i,j}1
 $$
for all $i,j=1,\ldots,n$, is of dimension $2^{2n}$ and is isomorphic to the algebra of $2^n\times 2^n$-matrices. In fact, one can do better: those matrices are matrices of all possible linear transformations of the Grassmann algebra $\Lambda(Y_1,\ldots,Y_n)$, on which $Y_i$ acts by multiplication, and $X_i$ acts as the <i>odd</i> derivation $\frac{\partial\phantom{Y_i}}{\partial Y_i}$. 

2. Since we are working with the matrix algebra, every module over it is a direct sum of several copies of the standard module. In particular, every module of dimension $2^n$ is isomorphic to the one given by the action on the Grassmann algebra. In particular, there is, up to rescaling, just one vector annihilated by all $X_i$.

3. All the questions in your post are addressed by this. Roughly speaking, you define a representation $\rho$ of this algebra by putting $\rho(X_i)=A_i$, $\rho(Y_i)=A_i^*$. The only thing to take care of is that you work over $\mathbb{C}$, not over $\mathbb{R}$, but this you can take care of by separating real and imaginary parts of your operators.