Can a solar flare reach far enough to travel from one ordinary star to another? If so, how can that solar flare affect the following star?
I'd say it's an open question awaiting some observational evidence that it can happen. It presumably would depend a lot on what kind of situation you are looking for, and what would you call a flare on one star setting off a flare on the other. Certainly you'd need two stars that both have active atmospheres and are in a close binary, such that one mass ejection could interfere enough with the magnetic configuration in the other star to set off a flare. Some stars may be so close together that the magnetic configuration of their atmospheres actually connect both stars, so then when get a flare you might see brightenings on both stars but it would be essentially the same flare. There are even stars called "contact binaries" where it's not even all that clear if you should think of that as two stars that touch (and share a magnetosphere), or just one star that has two separate cores. But the bottom line is, it sounds like you'd need something unusual going on, involving two stars that were unusually close together and are interacting in some way. You might want to read up on "RS CVn stars," for example.