Szeth got Vasher's sword from Nale. In the OB flashbacks we see The Nightwatcher offer Dalinar "a sword that bleeds darkness and cannot be defeated". We don't know how Nale came to have the sword, nor how The Nightwatcher got it nor how Vasher lost it.
Vivenna was presumably taught about Worldhopping from Vasher, who's been to Roshar before and in fact Nightblood was intended to emulate the Shardblades. We do know that she used Cultivation's Perpendicularity. A Perpendicularity is a hole, linking the Spiritual, Cognitive & Physical Realms, they are formed naturally when a high enough concentration of Investiture accumulates in one place.
Szeth doesn't have a normal Shardblade because he didn't complete the third Ideal till he reached Thaylen City. He was already using Nightblood & knew the sword was powerful, so probably chose to use that sword at that critical juncture. His Highspren might be a little frightened of Nightblood, I imagine. It's also possible that Szeth's third Ideal wasn't complete in a sense without Dalinar accepting it.