Well communication is certainly a key element in any poly relationship, but let's face it, some people are natural communictors; others have a hard time expressing themselves. You allow some leeway for the "communication challanged," without letting them off the hook. They still have to try to communicate.
Moving in together is certainly a huge step. Make sure all your ducks are in a row if you try it, and as much as it sucks to say it, have an emergency plan in case the poly relationship turns south and "K" turns out to be a real thorn in your side. You need to be prepared to effectively remove him from your premises if it comes to that. I'm just saying, once you hand someone a key to your house, you have handed them a heck of a thing. Far more complicated than if you were giving them a key to your car.
I am just speaking from personal experience. I let a person move in all too easily, and well it wasn't long before that person turned out to be someone completely different than how she presented herself in the beginning. My lady says, "Trust has to be earned," and I think that's especially true when it's your house you're talking about.
Other than that, it sounds like you have worked out most of your emotional issues so far, and are on even ground. Continue to do your best, and let us know how things are going.
Regards,
Kevin T.