Length of simple closed curve in half-translation surface

Yes, it is, of course, true. Even though foliations defined by $q$ are not orientable, the genus $2$ surface is orientable, so an $\varepsilon$-neighborhood of a closed geodesic $\gamma$ is just the cylinder $\gamma\times [-\varepsilon, \varepsilon]$.

This statement holds more generally for orientable surfaces with flat metric and conical singularities. If you find a simple closed geodesic (smooth one and disjoint from conical points) its $\varepsilon$-neighborhood is a cylinder.