Cook will get this series, and so will Trott. Michael Vaughan on TMS yesterday was saying that is the England way. They stick with what they know (hence Tredwell and Trott's selections) and give them a decent chance to come good. If Trott doesn't I think it's likely he'll be sent away to score runs and Lyth will get a go vs NZ.
Re; Cook I honestly feel the larger issue is the role of England captain. Hussain, Vaughan, Strauss and now Cook all found their batting suffering from being captain. The captain's role has meant much more off field requirements (PR, marketing, press and media, etc.), even more so in the lead up to test matches, especially at home. Got to keep the sponsors happy - they will all have demands for England player appearances and most will request the skipper.
That, in addition to his technical flaw, relationship with the head coach (which I don't believe is good - he clearly feels let down after being dropped) and the KP speculation and personal abuse has all taken a toll. As has been said, he's a proven test player and will be given, I think, the whole summer. He had a similar spell before his 700-odd runs in the ashes, but he wasn't captain then... If he isn't performing half way through the ashes, I suspect we'll see him stand down to concentrate on his batting...