Existence, uniqueness, and $L_p$-approximation results are presented for scalar stochastic differential equations (SDEs) by considering the case where, the drift coefficient has finitely many spatial discontinuities while both coefficients can grow superlinearly (in the space variable). These discontinuities are described by a piecewise local Lipschitz continuity and a piecewise monotone-type condition while the diffusion coefficient is assumed to be locally Lipschitz continuous and non-degenerate at the discontinuity points of the drift coefficient. Moreover, the superlinear nature of the coefficients is dictated by a suitable coercivity condition and a polynomial growth of the (local) Lipschitz constants of the coefficients. Existence and uniqueness of strong solutions of such SDEs are obtained. Furthermore, the classical $L_p$-error rate $1/2$, for a suitable range of values of $p$, is recovered for a tamed Euler scheme which is used for approximating these solutions. To the best of the authors' knowledge, these are the first existence, uniqueness and approximation results for this class of SDEs.