Not all swords can use Runic. Go into the inventory and view a weapons info to see what ability's it can be used with, extra's such as magic and elemental properties and whom can equip it.

For Zozo I take along Locke (Mandatory ofcourse), Edgar for his Auto-crossbow and Chainsaw, Celes for Cure as you'll likely to run out of Potions and the like, and Shadow for Shurikens as they are easily the biggest damage dealer at that point. If Shadow has hustled you and fucked off then a good replacement would be Sabin because of his Blitz skills such as Raging Fist and Fire Dance (?). As already been mentioned, have Locke Steal as much as possible especially from the Gigas' and any other enemies that hold Relics or other equioment.

Once you finally reach the top of the tower and embark on your next 'mission' you will have Locke and Celes mandatory. I don't know who's best to go with them story-wise so chose whomever.

Zozo is probably my least favorite city in any FF game. Usually I like getting into cities because I know that I am safe from random battles. Not in Zozo.

Though I've never had any trouble in this city. I think my characters may have been overleveled by the time I got there.

Did you pick up the Moogle's Charm? its hidden in the spot where Mog stood in the Moogle den.

Yeah 8 Dragon's to go through. The Wind Dragon is one of the harder ones mind I think. If you beat all of them you get the final Esper, Crusader.