1) Cubed items are account-wide, you can only extract every legendary once :-) You can, however, select different items on each character, so the current cube selection is not account-wide but per character.
2) They're around Torment 4 to 6. Some people perceive them much easier, and there are some dungeons that are harder, but not necessarily in terms of game difficulty but just in terms of mechanics. But officially, the HP/damage level of mobs is supposed to reflect T4-T6, and in some dungeons you also need to balance *your* damage/toughness to be on that level (to not kill mobs too fast). Set dungeons do not scale with game difficulty, it doesn't matter what difficulty your game is on.
3) You can level gems that are socketed, yes. It only matters if they're on your character (or in your stash). If you have a second character that has gems equipped, you won't see them when talking to Urshi. When you talk to Urshi, the first three gems you'll see are the ones you have currently equipped (assuming you have the maximum of three legendary gems equipped), and the other gems in Urshi's upgrade screen are the gems in your stash, in the order you have them organized in your stash.