1.) Uldah is in Thanalan. You can technically run there, but it's quite a trip especially without your chocobo. I would suggest playing through the story until you get access to the airship and you can travel to all the major cities.

2.) Yes, if you meet requirements for Duty Finder, it will place you into a suitable party (may take a while for anyone other than a tank).

3.) You need 30 Gladiator and 15 Conjuror for Paladin. Dragoon is 30 Lancer and 15 Marauder.
Also for your conveinance

Bard: Archer (30) -> Pugilist (15)
Paladin: Gladiator (30) -> Conjurer (15)
Dragoon: Lancer (30) -> Marauder (15)
Warrior: Marauder (30) -> Gladiator (15)
Monk: Pugilist (30) -> Lancer (15)
Scholar: Arcanist (30) -> Conjurer(15)
Summoner: Arcanist (30) -> Thaumaturge (15)
Black Mage: Thaumaturge (30) -> Archer (15)
White Mage: Conjurer (30) -> Arcanist (15)