Maximum distance separable (MDS) codes that are not equivalent to generalized Reed-Solomon (GRS) codes are called non-GRS MDS codes. Alongside near MDS (NMDS) codes, they are applicable in communication, cryptography, and storage systems. From theoretical perspective, it is particularly intriguing to investigate families of linear codes in which each element can be determined to be either a non-GRS MDS or an NMDS code. Two promising candidates for such families emerge from what is known as twisted GRS (TGRS) construction. These candidates are the $(+)$-TGRS codes and their extended versions, called $(+)$-extended TGRS (ETGRS) codes. Although many of their properties have been characterized, there are gaps to fill. Which among the codes are non-GRS MDS? Can we improve on their decoding by using their error-correcting pairs or deep holes? In this paper we solve these problems. The answer to the first problem leads us to two classes of non-GRS MDS Hermitian self-dual TGRS codes and a proof that there is no Galois self-dual ETGRS code. Addressing the second problem, we present an explicit decoding algorithm for ETGRS codes that outperforms existing decoding algorithms given some conditions. By considering the duals of TGRS codes which are MDS, we determine the covering radius and a class of deep holes of the recently constructed non-GRS MDS codes due to Han and Zhang.