As to your addendum regarding the fundamental group of Riemann surfaces, the situation is as follows: If $X$ is a smooth projective algebraic curve of genus $g$ over an algebraically closed field $K$ of characteristic $0$, and if $U\subset X$ is obtained by removing $r$ distinct closed points, then $\pi_1(U)$ is the profinite completion of the surface group \[\left<a_1,\ldots, a_g, b_1,\ldots, b_g, c_1,\ldots, c_r | [a_1,b_1]\cdot\ldots\cdot[a_g,b_g]c_1\cdot\ldots\cdot c_r = 1\right>.\] Phrased more traditionally in terms of Galois theory of fields, if $K$ is the function field of $X$, then this group is precisely the Galois group of the maximal algebraic extension of $K$ which is unramified with respect to all valuations of $K$ except those corresponding to the $r$ points that were removed.

More concretely: The finite coverings of $U$ arise by taking finite extensions of $K$, unramified except possibly at the removed points, and taking the normalization of $U$ in $L$ (that is, $U$ is an affine curve given by a ring $A$ contained in $K$, the covering will be the spectrum of the normal closure of $A$ in $L$).

As to your addendum regarding the fundamental group of Riemann surfaces, the situation is as follows: If $X$ is a smooth projective algebraic curve of genus $g$ over an algebraically closed field $K$ of characteristic $0$, and if $U\subset X$ is obtained by removing $r$ distinct closed points, then $\pi_1(U)$ is the profinite completion of the surface group \[\left<a_1,\ldots, a_g, b_1,\ldots, b_g, c_1,\ldots, c_r | [a_1,b_1]\cdot\ldots\cdot[a_g,b_g]c_1\cdot\ldots\cdot c_r = 1\right>.\] Phrased more traditionally in terms of Galois theory of fields, if $K$ is the function field of $X$, then this group is precisely the Galois group of the maximal algebraic extension of $K$ which is unramified with respect to all valuations of $K$ except those corresponding to the $r$ points that were removed.

More concretely: The finite coverings of $U$ arise by taking finite extensions of $K$, unramified except possibly at the removed points, and taking the normalization of $U$ in $L$ (that is, $U$ is an affine curve given by a ring $A$ contained in $K$, the covering will be the spectrum of the normal closure of $A$ in $L$).

Edit: If $K$ is not algebraically closed and $U$ is geometrically connected, then there is a short exact sequence \[1\rightarrow \pi_1(U\times_K \bar{K}) \rightarrow \pi_1(U)\rightarrow Gal(\bar{K}/K)\rightarrow 1\] (one has to pick compatible base points), and if $X$ has a $K$-rational point, then this sequence splits, so the structure is "known", as far as the Galois group of the base is known.