Beigel-Tarui transformation of ACC cricuits
Isn't the probability of $\Sigma_{i:h (i) =1}  x_{i} \mbox{mod }  2 = 1$ at least 1/2 ? It seems that $1/(10k)$ is a weak lower bound. 


In fact the answer is no. (It would be that $\Sigma_{i:h (i) =1}  x_{i} \mbox{mod }  2 = 1$ holds with probability at least $1/2-\varepsilon$, if we were working with an $\varepsilon$-biased hash family, and indeed using $\varepsilon$-biased hash functions gives a way to improve the parameters of the construction. But pairwise independence is not necessarily $\varepsilon$-biased.)

It seems they are missing one additional step here. To apply Valiant-Vazirani directly, you would need to also randomly choose the range of the hash function. Rather than picking random pairwise-independent $h : [2^k]\rightarrow\{0,1\}$, it seems you should pick random $\ell \in \{2,\ldots,k+1\}$ and then pick random pairwise-independent $h : [2^k]\rightarrow \{0,1\}^{\ell}$. (Here I am deliberately using Arora-Barak's statement of Valiant-Vazirani, found on page 354.) 
 Let $s$ be the number of $x_i=1$. Valiant-Vazirani says that when you have chosen $\ell$ such that $2^{\ell-2} \leq s \leq 2^{\ell-1}$, then the probability that $\Sigma_{i:h (i) =1}  x_{i} = 1$ (over the integers!) is at least $1/8$. 

So by picking random $\ell$ and picking random pairwise independent $h: [2^k]\rightarrow\{0,1\}^{\ell}$, then you have probability at least $1/(8k)$ that $\Sigma_{i:h (i) =1}  x_{i} \mbox{mod } 2 = 1$. To simulate the random choice of $\ell$ in the circuit, you could simply take the $OR$ over all possible $\ell$ (their number is logarithmic in $2^k$, after all), so the probability of success becomes at least $1/8$ again. So rather than having $O(k\log s)$ hash functions with range $\{0,1\}$, you'll want $O(k)$ different sets of hash functions (each set having a different range), with $O(\log s)$ hash functions in each set.


  Why does this replacement give an equivalent SYM+ circuit ?


A SYM of AND (i.e, SYM+) circuit of size $K$ is essentially equivalent to having a multivariate polynomial $h : \{0,1\}^n \rightarrow \{0,\ldots,K\}$ with at most $K$ monomials, a lookup table $g : \{0,\ldots,K\}\rightarrow \{0,1\}$, and computing $g(h(x_1,\ldots,x_n))$. (For instance, a proof can be found in Beigel-Tarui.) The intuition is that each monomial in $f$ is an AND gate, and $g$ is the SYM gate. I say "essentially equivalent" because the multilinear polynomial $h$ could also have negative coefficients for some terms, and negative coefficents are not obviously implementable in SYM of AND. But I claim (and Beigel and Tarui claim) that this is not a problem. Think about it :)