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We study numerically and analytically the dynamics of a single directed elastic string driven through a 3-dimensional disordered medium. In the quasistatic limit the string is super-rough in the driving direction, with roughness exponent $ζ_{\parallel} = 1.25\pm 0.01$, dynamic exponent $z_{\parallel}= 1.43 \pm 0.01$, correlation-length exponent $ν= 1.33 \pm 0.02$, depinning exponent $β= 0.24\pm 0.01$, and avalanche-size exponent $τ_{\parallel} = 1.09 \pm 0.03$. In the transverse direction we find $ζ_{\perp} = 0.5 \pm 0.01$, $z_{\perp} = 2.27 \pm 0.05$, and $τ_{\perp} =1.17\pm 0.06$. We show that transverse fluctuations do not alter the critical exponents in the driving direction, as predicted by the planar approximation (PA) proposed in 1996 by Ertas and Kardar (EK). We check the PA for the force-force correlator, comparing to the functional renormalization group and 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 2-dimensional random medium. While $z_{\perp}=z_{\parallel}+1/ν$ and $ν=1/(2-ζ_{\parallel})$ of EK are satisfied, the transversal movement is that of a Brownian, with a clock set locally by the forward movement. This implies $ζ_\perp = (2-d)/2$, distinct from EK. 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. In the fast-flow regime the local displacement distributions become Gaussian in both directions and the effective temperatures vanish as $T^{\perp}_{\tt eff}\sim 1/v$ and $T^{\parallel}_{\tt eff}\sim 1/v^3$ for RB and as $T^{\perp}_{\tt eff} \approx T^{\parallel}_{\tt eff} \sim 1/v$ for RF.