Regarding the mask, I think people forget that he is biologically related to Darth Vader.
The son of Han Solo AND Princess Leia, daughter of Vader. Not to mention that he was trained by the only known audience to Vader's Funeral, his uncle Luke Skywalker. We can assume that Leia pretty much hates Vader for what he had done, even if in Luke's eyes he's at least somewhat redeemed, and I feel like Han would be one to take trophies. Luke himself may have liked a reminder of Vader, to keep him on the right path to see what the consequences could be if he ever strayed off the right path (because it was pretty obvious that he was struggling by ROTJ).
So maybe as a family they kept it for whatever reasons they had.
Those factors combined could create a scenario where that remnant was taken from the pyre not by some recently turned Kylo Ren grave robber, and more of as an Heirloom stolen by an overly obsessed Ben Solo before or at the same time as becoming Kylo Ren. Given his direct connections, it is easily imagined that after slaughtering Luke's infant Jedi Academy he'd snatch the object of his obsession too as he high-tails out to go meet with sketchy Supreme Leader Snoke.