I think there's dozens of reasons they might move those guys.
They might think Hayward doesn't want to be in Boston. They might feel they need more size and he's the most movable piece. They might feel he's duplicative skill wise (or that he is unhappy in his role). They might feel they need more youth (although not likely!). They may not want to pay the tax down the road. They may feel he's going to leave next year and there's better options now. All plausible. None of us have any idea which is true, certainly including me. We just have ideas and preferences and beliefs among them which is fine.
On rumors, this is something that has been discussed on this board for more than a decade....really since it started. There are lots of things we see in the media. We know, from former front office execs who join the media (Hollinger and Marks being two who are now media) that many of the things in the media are flat-out false---teams never did them or thought about them. We also know that some are true, and some are partially true (yes, that was talked about, but only in the context of this other thing also happening, or yes, so and so's name came up but there were never any proposals) . And we also know (again, from guys in media and books written later on about teams) that lots of things that are discussed between teams are never leaked. So what we see in the media is never all that is being considered, and may not be real. Again, I enjoy seeing the rumors and discussing them...but I disagree with the idea that because it was in the media it must be so.
So is there relevance to something being in the media? Sure, some. Does it mean it is true? It simply does not. This is also where the reporter comes in---guys like Woj have a long track record and deep network of connections, so what they say has (in my mind---you can of course disagree) some real weight. They may be played, they may be sharing only part of a story, but they are (again, IMO) likely not just making things up. On the other hand, random bloggers do in fact just make things up (we have posters in this thread who are making up things about Kemba's knee!) and so some guy I've never heard of with no obvious media role or sources saying "Hayward and 14 for Turner" I give no greater weight than someone in this thread posting a fake trade. I in fact give it less---many of you are very knowledgeable and realistic and have shown that over the years---whereas I have no idea if that guy even knows what he is talking about (the fact his fake trade fails the cap is an indication he has no idea how this all works).
People can disagree with all of the above, of course....