We study numerically and analytically the dynamics of a single directed elastic string driven through a three-dimensional disordered medium. In the quasistatic limit the string is super-rough in the direction of the driving force, with roughness exponent ${\ensuremath{\zeta}}_{\ensuremath{\parallel}}=1.25\ifmmode\pm\else\textpm\fi{}0.01$, dynamic exponent ${z}_{\ensuremath{\parallel}}=1.43\ifmmode\pm\else\textpm\fi{}0.01$, correlation-length exponent $\ensuremath{\nu}=1.33\ifmmode\pm\else\textpm\fi{}0.02$, depinning exponent $\ensuremath{\beta}=0.24\ifmmode\pm\else\textpm\fi{}0.01$, and avalanche-size exponent ${\ensuremath{\tau}}_{\ensuremath{\parallel}}=1.09\ifmmode\pm\else\textpm\fi{}0.03$. In the transverse direction we find ${\ensuremath{\zeta}}_{\ensuremath{\perp}}=0.5\ifmmode\pm\else\textpm\fi{}0.01, {z}_{\ensuremath{\perp}}=2.27\ifmmode\pm\else\textpm\fi{}0.05$, and ${\ensuremath{\tau}}_{\ensuremath{\perp}}=1.17\ifmmode\pm\else\textpm\fi{}0.06$. Our results show that transverse fluctuations do not alter the critical exponents in the driving direction, as predicted by the planar approximation (PA) proposed by Ertas and Kardar (EK) [Phys. Rev. B 53, 3520 (1996)]. We check the PA for the measured force-force correlator, comparing to the functional renormalization-group and numerical simulations. Both random-bond (RB) and random-field (RF) disorder yield a single universality class, indistinguishable from the one of an elastic string in a two-dimensional random medium. While relations ${z}_{\ensuremath{\perp}}={z}_{\ensuremath{\parallel}}+1/\ensuremath{\nu}$ and $\ensuremath{\nu}=1/(2\ensuremath{-}{\ensuremath{\zeta}}_{\ensuremath{\parallel}})$ of EK are satisfied, the transversal movement is that of a Brownian, with a clock set locally by the forward movement. This implies ${\ensuremath{\zeta}}_{\ensuremath{\perp}}=(2\ensuremath{-}d)/2$, distinct from EK. Finally, at small driving velocities the distribution of local parallel displacements has a negative skewness, while in the transverse direction it is a Gaussian. For large scales, the system can be described by anisotropic effective temperatures defined from generalized fluctuation-dissipation relations. In the fast-flow regime the local displacement distributions become Gaussian in both directions and the effective temperatures vanish as ${T}_{\mathtt{eff}}^{\ensuremath{\perp}}\ensuremath{\sim}1/v$ and ${T}_{\mathtt{eff}}^{\ensuremath{\parallel}}\ensuremath{\sim}1/{v}^{3}$ for RB disorder and as ${T}_{\mathtt{eff}}^{\ensuremath{\perp}}\ensuremath{\approx}{T}_{\mathtt{eff}}^{\ensuremath{\parallel}}\ensuremath{\sim}1/v$ for RF disorder.