Does every connected, compact Riemann surface $\Sigma$ with boundary, $\partial \Sigma\not =\emptyset$, admit a holomorphic function (smooth on the boundary) $f:\Sigma\to\mathbb C$ whose derivative is everywhere non-zero? 



Does every Riemann surface with boundary immerse in C?
A more general result is proven in 


Gunning, R. C., Narasimhan, R., Immersion of open Riemann surfaces. Math. Ann. 174, 103–108 (1967).


As for compact surfaces with boundary, it is essentially a part of the definition that they embed holomorphically into open Riemann surfaces. 


Incidentally, it is an open problem for $n$-dimensional Stein manifolds (with trivial complex tangent bundles), $n\ge 2$, if they admit locally biholomorphic immersions in ${\mathbb C}^n$. The best (to my knowledge) result in this direction is due F.Forstenič (Acta Math., 2003) who proved that every complex parallelizable $n$-dimensional Stein manifold admits a holomorphic submersion to ${\mathbb C}^{n-1}$. 


