Actually I'm 100% sure they will leave shadowmourne obtainable. There is a guild achievement where you need each legendary in your guild. If they remove shadowmourne, it means that this achievement is no longer available and would then be a feat of strengh. Recruiting a wielder of shadowmourne is not the way blizz intended this achievement to be obtained (because yes I know, Blizz said that the legendaries would count retroactively for this achievement)
And shadowmourne is worse than any lvl 85 blue weapon so no balance issue either.
I would assume they will stay since we haven't heard anything about it.
Pretty sure you're wrong. The BoE chest mount isn't a drop. The chest is a drop. If they remove the chest, they remove that mount. Making that chest a rare drop wouldn't suffice to accomplish anything, as once it drops it drops for everyone in the raid with a Shadowmourne that hasn't completed the quest. As well as the quest reaching completion when you kill the LK, so that would have 1 in every 100 Shadowmournes getting the chest? Yeah, that would be a solution.