A few questions about this game.

[Jeeqbit]

You can use the same characters, but you have to buy the PS4 version of the game separately to play it on PS4, then enter the code into your existing account.

There is a backup option on the character selection screen which backs up your character or system settings to the cloud, so you can restore them the same way on another device. The button at the top backs up your general settings and the button next to each character backs up the settings for that character, so you should do both.

The game supports controllers really well and I don't know about extra buttons, but I do know that people play with an MMO mouse so there are ways to do it.

You can copy your character settings from one character to another. Go to Documents\My Games\Final Fantasy XIV - A Realm Reborn and look for your CHAR folder. The one that contains a log folder with chat logs that were modified today is most likely the character you have been using, so you can copy the HUD file and paste it into the other CHAR folders.

It's wise to backup the Final Fantasy XIV - A Realm Reborn folder before you do this by copy and pasting it to the desktop, in case you do it wrong.