Elgamal problem on $\mathbb{QR}_p$ with $p$ a safe prime
$P[s(g^{ab}) = i] = \frac{q-1}{q^2}$ if $i \neq 1$ and $\frac{2q-1}{q^2}$ if $i = 1$. 


We assume that $g$ is a generator of $\mathbb{QR}_p = \mathbb Z_q$. Then, as you said, in order to study the probability that $g^{ab} = i = g^c$ for a fixed $c$, we can calculate the probability that $ab = c\bmod q$. 

In general, the probability that a random pair $(a,b)\in\mathbb Z_q\times\mathbb Z_q$ gives you a fixed non-zero number $c$ is $(q-1)/q^2$, and if $c=0$ then this probability is $(2q-1)/q^2$.


  2) I don't currently see how to take $s$ into play here.


ElGamal encryption scheme allows you to encrypt elements $m\in\mathbb{QR}_p$ as $m\cdot g^{ab}$. However, the message you want to encrypt is likely not to be an element of this set.
In practice, you encrypt bit-strings, which are easier to map into $\{1,\ldots,(p-1)/2\}$. Then you take an element $y$ in this interval and map it to $m\in\mathbb{QR}_p$ via $m = y^2$, and use $m$ as above.
Now you can see that you need $s$ for decryption in order to recover $y$ from $m$.


  3) I believe the reasoning is equal to point 1).


Agreed. The idea is that with a safe prime $p = 2q+1$ you know that $\mathbb{QR}_p = \mathbb{Z}_q$, whereas in general this structure could be more involved.
However, once you're working over $\mathbb{Z}_q$, the analysis is the same.