To evaluate that limit, we can expand each function in a Laurent series at $s=0$ and use the following 3 facts about the Hurwitz zeta function:
$$ \zeta(-s,a) = \zeta(-s,a+1) + a^{s} \tag{1}$$
$$ \zeta'(-s,a) = \zeta'(-s,a+1) -a^{s} \log(a) $$
$$\zeta(-n, a) = -\frac{B_{n+1}(a)}{n+1} \ , \ n \in\mathbb{N} \tag{2}$$
Doing so, we get
$$z - z \log z - \frac{\gamma z^{2}}{2} + \lim_{s \to 0^{+}} \Big[ - \Gamma(s-1) \zeta(s-1,z+1) -z \Gamma(s) \zeta(s) + \frac{z^{2}}{2} \Gamma(s+1) \zeta(s+1)$$
$$+ \Gamma(s-1) \zeta(s-1) \Big]$$
$$ = z - z \log z - \frac{\gamma z^{2}}{2}$$
$$ + \lim_{s \to 0^{+}} \Bigg[-\Big(-\frac{1}{s} + \gamma -1 + \mathcal{O}(s) \Big) \Big( -\frac{z^{2}}{2}+\frac{z}{2}-\frac{1}{12}-z + \zeta'(-1,z)s + z \log z \ s + \mathcal{O}(s^{2}) \Big)$$
$$ - z \Big( \frac{1}{s} - \gamma + \mathcal{O}(s) \Big) \Big( - \frac{1}{2} - \frac{\log (2 \pi)}{2} s + \mathcal{O}(s^{2}) \Big) + \frac{z^{2}}{2} \Big(1- \gamma s + \mathcal{O}(s^{2}) \Big) \Big( \frac{1}{s} + \gamma + \mathcal{O} (s) \Big) $$
$$ + \Big(- \frac{1}{s} + \gamma -1 + \mathcal{O} (s) \Big) \Big( - \frac{1}{12} + \zeta'(-1) s + \mathcal{O}(s^{2}) \Big) \Bigg] $$
$$ = z - z \log z - \frac{\gamma z^{2}}{2}$$
$$ + \lim_{s \to 0^{+}} \Big[\zeta'(-1,z) + z \log z + \frac{\gamma z^{2}}{2} - \frac{z^{2}}{2} + \frac{z}{2} - z + \frac{z \log(2 \pi)}{2} - \zeta(-1)+ \mathcal{O}(s) \Big] $$
$$ = \frac{z}{2} \log(2 \pi) + \frac{z(1-z)}{2} -\zeta'(-1)+ \zeta'(-1,z) $$
$ $
$(1)$ http://dlmf.nist.gov/25.11 (25.11.3)
$(2)$ http://mathworld.wolfram.com/HurwitzZetaFunction.html (9)
Best Answer
One trivial mistake that blows your argument away is that RZ is infinity at 1 so its log is not defined in your S; you need S to exclude the real line, so you have to work for t>T>0 as otherwise, you cannot apply the residue theorem since you would have a logarithmic singularity at 1
(edited later after tons of corrections by the original poster to address various objections for careless flaws that appeared in the original computation):
The main flaw of the computation lies in the misunderstanding of the complex-log, namely that while the complex-log behaves on branch points as noted in the paper by Bui et al referred in the comments from which the original poster copy pasted his argument - so it jumps by 2*Pii when crossing such-, its primitive doesn't as shown by the simple example of log(z) vs zlog(z)-z on the complex plane minus the negative axis.