unitron

The Watch takes no part, but the flip side of that deal is that while the noble houses and knights and lords and ladies may squabble amongst themselves, they're supposed to leave the Watch out of it, and further they're supposed to all be helping support the Watch. Out of all of those squabbling contenders for the Iron Throne, the only one who could be bothered to do any more for the Watch than to tell them to eff themselves was Stannis. So Jon bends the rules a little, lets Stannis take some old spears and junk armor which he could have taken by force if he'd been of a mind to, gives Stannis a little Northerly advice, because otherwise he's letting his only ally against the others ride out to be slaughtered by the Boltons. And then comes the Pink Letter, which personally threatens Jon and more generally threatens the Watch as well. So to protect the sworn brothers of the Watch Jon detaches himself to ride south without them to intercept the Boltons to keep them away from the Watch and The Wall. It just so happens there're are a bunch of non-sworn wildlings who accept his invitation to ride along and keep him company. Jon wasn't going to retake Winterfell, he was going to intercept the Boltons so as to keep all the rest of the Watch but himself out of the conflict. Winterfell is no longer his home.

I think they accept them after they and their alleged crimes have been dealt with by the proper authorities and they have been given the option of taking The Black instead of whatever punishment would have been dealt out to them otherwise. And of course for noble houses looking to dispose of a third or fourth son they can't marry off advantageously, The Watch will gladly accept "volunteers".

There is no way for the Lord Commander of the Watch to execute someone for acts not committed on Watch land without it being "taking a part", which they don't do.* Though they might hold them for extradition back to the area of the crime. *(or at least aren't supposed to, although Jon Snow has kinda been forced into bending that rule a bunch by circumstances largely beyond his control).

If, as you suggest, Roose rides out leaving Ramsay "holding the fort", then Roose has the House Bolton seal with him, so Ramsay can't use it to put an impression in the pink wax, and for that matter, may have had to salvage/scavenge that smear of pink wax by melting it off of some other document he still had lying around, so the lack of the Bolton seal impressed into the wax isn't necessarily an indication that it wasn't Ramsay who wrote the Pink Letter.

As Lord Commander of the Watch, Jon has no authority to execute Theon, unless Theon joins the Watch, and then, ala Slynt, refuses a direct order.

More likely a reference to the last days of Howard Hughes, who was also noted for a descent into poor mental health in his declining years.