We have several interior doors that only stick or won't completely close in the winter. We also get cracks in the drywall mud above the doors so I think something else is going on besides just humidity changes in the door and frame. This is a wood framed house with wood trusses and a metal standing seam old-fashioned roof. I remember hearing a DIY radio show discussing the issue but I can't remember the solution. I think it had something to do with the wooden trusses in the attic. Any information is appreciated.
Since you say the doors only stick in the winter, that would lean more towards the humidity/temp changes you mention. As for the cracks, could be part of that, could be from just natural settling of the home over time.
As to things to do about it, I'd just point you to an easily found article on the net where someone else has already gone through the trouble.
Hope this helps