I never subscribed to the idea that Luke sent Kylo to infiltrate Snoke's ranks. He has every reason to take Snoke out- he's leading the First Order (to a degree) and he's a dark force user, but I feel like that's too zealous and aggressive for Luke. I also doubt he would ever put his nephew in harm's way like that. I think it's more reasonable that Kylo had wanderlust while training, and since Luke was kept in tight rein on the homestead on Tatooine he didn't want Kylo to feel the same desire to leave that he had. I don't think he ever expected another Sith-like entity to sweep him up like that, as he had already eliminated what he thought were the last in the galaxy.
I've been assuming that the Knights of Ren are most of Luke's previous students. Since the movie seems to reflect ANH so much, I thought it would be a nice touch if Kylo 'killed' most of Luke's students in the same way that Vader 'killed' Luke's father.
In terms of Kylo trying to pressure Rey into learning from him, I assumed it was him trying to follow in Vader's footsteps of recruiting a powerful apprentice to help overthrow his master; not to redeem himself, but to rule the dark side. Add to that the fact that Rey may be a Skywalker and you've got a nice allusion soup going there. There is also a layer of added irony in that, in trying to convince Rey to give in to his will, he teaches her more about how to be a Jedi than anyone else. She learns how to force persuade people from Kylo's attempts to invade her mind. In Kylo's furious attack in the end, she finally sees Maz Kanata's wisdom in opening herself up tot he force. And if not for her desperation to save Finn in the end she may not have learned how to use Force Pull so soon.
While I'm on the topic, we know that Luke struggled to learn to use the force in between Empire and ANH from the new canon materials that take place between them. It makes comparatively little sense for Rey to do all of the things she does instinctively, in that case. From this perspective, I believe that Rey was at least one of Luke's former students, who either underwent enough psychological trauma or the good ol' Revan treatment to suppress her memories. Luke, wanting to save at least one of his students, sent her to a remote world under the watch of someone he trusts (Max von Sydow's character) until she was ready. In this case he also believed that Rey would eventually be drawn to him, which is also why he placed the map fragment that would lead to his location on Jakku- for Rey to find or be given. This would also be a call back to ANH, since the Jedi Masters know their apprentice is out there, but will not take action until their future padawan is called to action by the force.
The best part about that is that Rey doesn't necessarily need to be a Skywalker for it all to work out, though it does add a layer of depth.