Why is the Pokémon Refresh button shaking

You can't trade Pokémon to/from Pokémon Sun and Moon with any other games.

Eventually, however, you will be able to transfer Pokémon to Sun or Moon from ORAS, XY, BW, BW2, and the virtual console RBY versions.

See here:

Pokemon Sun and Moon are not compatible with the Generation 6 series of games: Pokemon X and Y; and Pokemon Omega Ruby and Alpha Sapphire. You won't be able to trade or battle between these games and Pokemon Sun and Moon, therefore you'll need a special app called the Pokemon Bank to transfer your Pokemon.

Transporting Pokemon into Sun and Moon from Pokemon White, White 2, Black, Black 2 and the Virtual Console games Pokemon Red, Blue, and Yellow is also possible with Poke Transporter, which comes bundled with Pokemon Bank.

However, Pokémon Bank and Poké Transporter won't be available for Pokémon Sun or Moon until January 2017.

Directly quoting someone who's read the compiled code, IVs, shininess, and hidden abilities are affected as follows:

If chain is less then 5 -> Fixed 31 IV Count = 0
If chain is less then 10 -> Fixed 31 IV Count = 1
If chain is less then 20 -> Fixed 31 IV Count = 2
If chain is less then 30 -> Fixed 31 IV Count = 3
If chain is 30 or more -> Fixed 31 IV Count = 4

If chain is less then 10 -> Hidden Ability Chance = 0%
If chain is less then 20 -> Hidden Ability Chance = 5%
If chain is less then 30 -> Hidden Ability Chance = 10%
If chain is 30 or more -> Hidden Ability Chance = 15%

About the shiny chance, well the code is not easy to read on this, but it does not affect the chance, it affects the try count.
Everytime a pokemon is generated, there is a "Shiny try count", by default it's 0, so the functions gives up after 1 try.
But somehow the chain count affects this try count, the min is 0, the max is 3. If i read the code correctly, then it starts after a chain of 70.

Note: this means, it's the default 1/4096 for each try, if 3 trys it's 4/4096 -> 1/1024.

It's not clear exactly what they were trying to say happens at a 70-chain for shiny chances - whether the chances max out at a 70 chain, or whether a 70 chain is the first point at which shiny chances increase, or what. We're already seeing people making conflicting assumptions about that. It's also unclear how this interacts with the Shiny Charm.

There's been at least one report of the chain counter overflowing back to 0 after hitting 255, resetting the above benefits. I've only seen one report, and no one who's read the code seems to have confirmed it, so it could be wrong.

Additionally, SOS chaining also provides improved EV gains. /u/foxhull on Reddit claims to have tested and found that all EV gains from an SOS battle are doubled, including EVs from the original wild Pokemon and from power items, stacking multiplicatively with Pokerus. Other people in the thread seem to agree.

SOS chaining does not boost your chances of getting a 5-IV or 6-IV.