Are you sure you are plugging it correctly? If so, you might have smashed the connectors whether on the screen or the logicboard, they are very sensible. Remember not to hold the screen in 90º with the case, or you will damage the flexs. Do you have another iPhone 6 where you can try on the LCD? Let us know how it goes!
For what you say I would say that the issue comes from the logicboard because there is more than one system that is failing: LCD and Sim Card. Do you have any spare iPhone 5 where you can test the simcard/LCD? - That would be a good start; but for what you say, my first thought is: logicboard. Let us know how it goes! P.S: logicboards are not cheap, I've seen some for under $100.00
