Let q=ph be a prime power and e be an integer with 0≤e≤h−1. e-Galois self-orthogonal codes are generalizations of Euclidean self-orthogonal codes (e=0) and Hermitian self-orthogonal codes (e=h2 and h is even). In this paper, we propose two general methods to construct e-Galois self-orthogonal (extended) generalized Reed-Solomon (GRS) codes. As a consequence, eight new classes of e-Galois self-orthogonal (extended) GRS codes with odd q and 2e|h are obtained. Based on the Galois dual of a code, we also study its punctured and shortened codes. As applications, new e′-Galois self-orthogonal maximum distance separable (MDS) codes for all possible e′ satisfying 0≤e′≤h−1, new e-Galois self-orthogonal MDS codes via the shortened codes, and new MDS codes with prescribed dimensional e-Galois hull via the punctured codes are derived. Moreover, some new q-ary quantum MDS codes with length greater than q+1 and minimum distance greater than q2+1 are obtained.