Stable resolution of a $2\times2$ linear system Cramer's method for the resolution of linear systems is known to be unstable, even in the $2\times2$ case. For general systems, stability can be improved by partial or full pivoting.
When you transpose the full pivoting principle to a $2\times2$, the procedure essentially amounts to


*

*finding the LHS coefficient with the largest magnitude, let it be $a_{11}$ WLOG;

*computing $x_2$ by determinants*,

*computing $x_1$ by elimination of $x_2$ from equation $1$.
Can this improve stability ? Is there a more stable solution ?

*Whatever the choice of the pivot, the formula amounts to a ratio of $2\times2$ determinants. I wonder if first normalizing the pivot coefficient to $1$ makes any difference.
$$\frac{b_1-b_2\dfrac{a_{21}}{a_{11}}}{a_{22}-a_{12}\dfrac{a_{21}}{a_{11}}}\text{ vs. }\frac{b_1a_{11}-b_2a_{21}}{a_{11}a_{22}-a_{12}a_{21}}$$
 A: If your system has a fused multiply add instruction, then most determinants $$x = ab-bc$$ can be evaluated accurately using Kahan's method
$$\hat{w} = \text{fl}(bc), \quad e = \text{fl}(\hat{w} - bc), \quad \hat{f} = \text{fl}(ad-\hat{w}), \quad \hat{x} = \text{fl}(\hat{f}+e).$$
Here $\text{fl}(x)$ denotes the floating point representation of $x$. In the absence of underflow or overflow, we have the relative error bound $$|x - \hat{x}| \leq 2 u |x|.$$
Here $u$ is the unit roundoff. This is far better than we have any right to expect. Baring floating point exceptions you can solve a 2 by 2 linear system with a componentwise forward relative error which is at most $$\gamma_4 = \frac{4u}{1 - 4u}.$$
The proof of these and related statements are contained in the paper. 
Further analysis of Kahan's algorithm for the accurate computation of 2-by-2 determinants
If your system does not have fused multiply add instruction, then I would try to combine the regular TwoProduct and TwoSum algorithms, see the paper
Error-free transformations in real and complex floating point arithmetic 
for a description of the these and other algorithms based on error-free transformations of basic arithmetic operations.
In truth, I have not completed the analysis, but I would be very surprised if this idea did not work. If I stumble, then I would fall back on the paper 
Accurate sum and dot product
and treat the determinants as very short inner products.
Finally, the battle against overflow or underflow can be won by extracting and manipulating the exponents on your own.
A: I have not a scoop, but it won't do any harm.
Using partial or full pivoting allows to benefit from the minimum error associated to the condition number $A$, $CN(A)\approx 10^r$. If we perform the previous calculations with $k$ significant digits, then we obtain the solution with $k-r$ significant digits. With Reduced Row Echelon Form, we cannot hope a better approximation. If we choose arbitrary $A,b$, then, roughly speaking, $A$ is diagonalizable, $CN(A)\approx \rho(A)\rho(A^{-1})$ (eventually with a factor $10$) and $b$ has always a non-zero component on all the $A$-eigenvectors.
Along my tests with $n=2$, the iteration methods do not improve the accuracy of the obtained solution (obviously when we fix the number of significative digits).
EDIT. About your last question, we lose significant digits during subtractions and not during divisions; therefore, in my opinion, the $2$ proposed methods have same accuracy.
