1. Yes, you can now drag your pk6 files to your S/M game.
2. PKHeX does the same that bank does when you transfer them.
3. There's no ingame national dex in Sun and Moon.
Lord, can you get the Galileo (i guess this was the OT or something) Rayquaza wondercard, which is shiny, drag it to PKHeX and just change the moves? I think that's not difficult to do.
That's an issue that has been fixed already as per this commit https://github.com/kwsch/PKHeX/commit/b7a29c1017db2db57ef16079c99e30d4c37b0cbc
And well, I don't know if you can use past gen7 Pokémon (unless hatched from egg) in battles online.