Let $\Sigma_{g,n}$ be a closed genus $g$ Riemann surface with $n$ marked points. $\Sigma_{g,n}$ has Fuchsian as well as Schottky uniformization. Fuchsian uniformization is done by a Fuchsian group and Schottky uniformization is done by a Schottky group. Then:
- What is the number of generators of the Fuchsian group in terms of $g$ and $n$?
- What is the number of generators of the Schottky group in terms of $g$ and $n$?
- Let $\{\gamma_a^F\}$ be a set of Fuchsian generators and $\{\gamma_a^S\}$ be a set of Schottky generators. How can these two sets be related to each other? In other words, given $\{\gamma_a^F\}$, how can one construct $\{\gamma_a^S\}$ or given $\{\gamma_a^F\}$, how can one construct $\{\gamma_a^S\}$? Is there any systematic way?
Your question seems to be still ill-defined. First of all, while there is a standard definition of Schottky uniformization for compact surfaces, there does not seem to be one for punctured surfaces. Here is an ad-hoc definition. In particular, I see no meaningful way to define it when the number of punctures is odd. Therefore, I will assume from now on that $n=2p$ is even. I will also assume that either $g> 1$, or $g=1$ and $n\ge 2$, or $g=0$ and $n\ge 4$.
Definition. A Schottky-type group of type $(g,p)$ is a free discrete subgroup $G$ of $PSL(2, {\mathbb C})$ which is free on $g+p$ generators, where $g$ of the free generators (namely $A_1,...,A_g$) are loxodromic and $p$ generators (namely $C_1,...,C_p$) are parabolic, and the limit set of $G$ is homeomorphic to a Cantor set.
One can show that if $\Omega$ is the domain of discontinuity of $G$ then the quotient surface $\Omega/G$ has genus $g$ and $2p$ punctures. This answers your question 2.
For question 1, you do not need the requirement that $n$ is even. You have to specify what a "Fuchsian generating set" is. Again, for closed surfaces, one can say that this is a standard generating set of the compact surface of genus $g$, equivalently, any generating set consisting of $2g$ elements. When you allow $n>0$ punctures, I guess, you can say that this is a set of free generators $a_1,b_1,...,a_g, b_g, c_1,...,c_{n-1}$ consisting of $2g+(n-1)$ elements such that $$ c_1\ldots c_{n-1} c_n \prod_{i=1}^g [a_i,b_i] = 1, $$ where each $c_i$ is represented by a loop going once around the $i$-th puncture (in the positive direction with respect to the orientation of the surface).
As for your question 3, my guess is that the relation is given by a homomorphism $b_k\to 1$, $a_j\to A_j$, $c_{i} c_{p+i} \to C_i$ for each $i, j, k$.
As for references, maybe this paper and references therein would be useful. (I got it by googling.) Another possible references is the book "Kleinian Groups" by B.Maskit.