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Is it always possible to give a complex structure on the one-point compactification of a Riemann surface such that compactification will be a Riemann surface?
If it is possible, can we extend a proper holomorphic map between two Riemann surfaces to their corresponding one point compactifications holomorphically?
No, this is not always possible. Indeed, the one-point compactification of a Riemann surface may not even be a manifold. For instance, the one-point compactification of $\mathbb{C}\setminus\{0\}$ is a sphere with two points identified, which is not locally homeomorphic to $\mathbb{R}^2$ at that point.
If $S$ and $T$ are Riemann surfaces which happen to have one-point compactifications $S^*$ and $T^*$ which are Riemann surfaces, then any proper holomorphic map $f:S\to T$ does extend holomorphically to a map $S^*\to T^*$. By properness, we can extend $f$ continuously to a map $S^*\to T^*$, and this map is holomorphic at $\infty$ since the singularity must be removable (in coordinate charts at $\infty$, $f$ is bounded near the singularity).
