I'm also new to the subject, but I'll try to share what I've learned so far if you haven't found the answers already. First, since we want to introduce a charged Black Hole in the bulk ($Reissner$-$Nordstr\ddot{o}m$-$AdS$ Black Hole), it'll be dual to a field theory at finite temperature and charge density with the partition function of the grand canonical ensemble:
$$\mathcal{Z}=tr\Big[\exp \Big(-\beta \,(\hat{H}-\sum\limits_{i}{{{\mu }_{i}}{{{\hat{Q}}}_{i}}})\Big)\Big]$$
where $\beta$ is the inverse temperature and ${\mu}_{i}$ are the chemical potentials associated with the charges ${\hat{Q}}_{i}$ which themselves are the Noether charges associated with conserved currents $J_{(i)}^{\nu }$ under global $U(1)$ symmetry. For $\mathcal{N}=4\,$ SYM, these currents are conserved under the $U(1)$ subgroup of the global $SO(6)$ R-symmetry [1][2].
Now for simplicity, first let's consider the pure $Ad{{S}_{5}}$ with a non-zero electric local gauge field ${{A}_{0}}(z)$ , where the index is for its temporal component. Solving the equations of motion will yield an asymptotic solution of the form:
$${{A}_{0}}\simeq A_{0}^{(0)}(1+A_{0}^{(1)}{{z}^{2}}) \quad as \quad z\to0$$
By the AdS/CFT dictionary, since the partition function of the field theory will have the additional source term of the form $\int{{{d}^{4}}x}\,{{J}^{\mu}}{{A}_{\mu}}$ , therefore the boundary value of the bulk gauge field (i.e. $A_{0}^{(0)}$) will be the source (i.e. chemical potential) conjugate to the conserved charge density $J^0$ [3].
Now by keeping in mind the above reasoning, in the case of $Reissner$-$Nordstr\ddot{o}m$-$AdS$ BH we will have the following form for the bulk gauge field:
$${{A}_{0}}\propto Q\,(z_{h}^{d-2}-{{z}^{d-2}})\quad for \quad d\ge 3$$
where $z_{h}$ is the location of the event horizon obtained by $f(z_{h})=0$ . You can find the explicit form of $f(z)$ in [4]. Now recall that the chemical potential is the boundary value of the gauge field, so we'll have:
$$\mu =\frac{1}{L}\underset{z\to 0}{\mathop{\lim }}\,{{A}_{0}}(z)$$
where $\frac{1}{L}$ is introduced to ensure that $\mu$ has the correct dimension of energy in the boundary theory. For your 3rd question, by computing the Hawking temperature (T), the ratio $\frac{\mu}{T}$ will give us a good transition from zero charge where $\frac{\mu}{T}=0$ to the extremal case where $\frac{\mu}{T} \to \infty$ for the fixed $z_{h}$. You can find its explicit form and the parameters it depends, with some good discussions in [4]. I hope this answer will be helpful.
[1] J. Kapusta, et al., Phys. Rev. D 28, 3093 (1983)
[2] H. Nastase, Introduction to the AdS/CFT Correspondence, CUP, 2015
[3] M. Natsuume, AdS/CFT Duality User Guide, Springer, 2015
[4] D. Galante, M. Schvellinger, JHEP 1207 (2012) 096, https://arxiv.org/abs/1205.1548 (2012)
