If you need to heal up, change your characters into White Mages, because they have the lowest HP in the game. You can use regular Potions or low level white magic spells to cure them. Revert them back to their original classes and they'll be at max health.

Fulfill these requirements to open up extra dungeons.

Chocobo Dungeon: Talk to Clasko after you raise at least 3 Chocobos to Level 5.
Fiend Colony: Raise a Level 5 Chocobo and send it to Mihen Road.
Via Infinito: In the room with the lift, in Chapter Five, there is a group of Kinderguardians, and a portal by them.

To reset in-game, press all four shoulder buttons (L1+L2+R1+R2), START and SELECT.

During the dance mini-game, ignore the commands of the game and simply press X + SQUARE + TRIANGLE + CIRCLE repeatedly to earn 500 points or more.

Complete the game once through. Afterwards, there should be a "New Game +" option on the title screen that lets you restart the game with all the items your characters had before completing the game the first time.