I do not believe this is correct. I would have to check the CBA to verify this. My impression has always been that if he actually signs the tender, he is then considered under contract with the Patriots, which limits his ability to talk freely with other teams, receive formal offer sheets, visit facilities, etc.
Yes we have already established several days ago that officially this is correct, but unofficially a sign-and-trade is what, for all intents and purposes, could happen (similar to the Welker trade).
Of course they can. You are not allowed to tender a player in bad faith for the purposes of restricting their movement, but NE and NO most certainly can discuss hypotheticals such as "would you take the #32 for Butler if we come to a separate contract agreement with him?"