sorry if this is not long enough but basically tay is his twin brothers luck bringer, so all of his luck is from tay, and because jay knows this he wanted to understand how it would we like to have his own luck without tay being there so that's why he left and that's why he had been captured and been in bad experiences because he dint have tay with him
Jaeui severed their "red string of fate" but it didn't stop Taeui from being his gilsangcheon. Taeui still brings Jaeui luck, but in return now Taeui suffers from bad luck. Before when they were still connected, it was almost a mutual relationship, where Jaeui still gets his good luck from Taeui, and Taeui lives a normal life.
Jaeui will always get his good luck from Taeui as long as Taeui is ALIVE. If Taeui dies, Jaeui loses his good luck and would die quickly bc of his dangerous work. So you are right that Jaeui getting Rahman is due to his good luck.
And right now, Taeui's red string of fate has been connected to Ilay. That's why Ilay manages to find Taeui all the time and why Ilay is able to meet up with him when he wants to.

Sorry guys but someone could explain for me this luck thing and the mysterious relationship of the twins? I feel like I'm not understanding the story deep enough