Let $q=p^m$ be a prime power and $e$ be an integer with $0\leq e\leq m-1$. $e$-Galois self-dual codes are generalizations of Euclidean $(e=0)$ and Hermitian ($e=\frac{m}{2}$ with even $m$) self-dual codes. In this paper, for a linear code $\C$ and a nonzero vector $\bm{u}\in \F_q^n$, we give a sufficient and necessary condition for the dual extended code $\underline{\C}[\bm{u}]$ of $\C$ to be $e$-Galois self-orthogonal. From this, a new systematic approach is proposed to prove the existence of $e$-Galois self-dual codes. By this method, we prove that $e$-Galois self-dual (extended) generalized Reed-Solomon (GRS) codes of length $n>\min\{p^e+1,p^{m-e}+1\}$ do not exist, where $1\leq e\leq m-1$. Moreover, based on the non-GRS properties of twisted GRS (TGRS) codes, we show that in many cases $e$-Galois self-dual (extended) TGRS codes do not exist. Furthermore, we present a sufficient and necessary condition for $(\ast)$-TGRS codes to be Hermitian self-dual, and then construct several new classes of Hermitian self-dual $(+)$-TGRS and $(\ast)$-TGRS codes.