Jump just determines the maximum height you can ascend. With high enough jump you can pretty much jump up any size cliff. I've heard of people getting stuck in dungeons when stairs get placed on chimneys or something, but I've never had that happen to me.
You could probably breeze through the balloon dungeon gimmick (which returns in a dengeki dungeon and the absolute hero mastery dungeon...) with high jump.
I honestly don't know about costume stats. Supposedly, costumes grant some stat bonuses/minuses. I think they modify the stats you gain while leveling, because I can't see any difference at level 1 and they're not documented ingame. I would guess that Laharl would give you more SAT as you leveled, but I haven't tested it so I dunno. The only things you can see are the changes in elemental affinity; Index has 50% to all elements, Etna has 50% wind, etc etc.
The stats for the workers are used for the facilities they work at.
Home - Love
Blacksmith - Smithing
Caravan - Battle, Transport
Insurer - Info
Church - Love, Healing
Dark Clinic - Healing
Delivery - Transport
Tiger's Den - All
COSPATIO - Transport
Space Police - Battle
Usual Cafe - Transport
Satellite Cannon - Info, Smithing
Raising the stats allows you to call the facility more times while in a dungeon, insurer protects more items, and the church can be set higher. Some of the facilities don't seem to do anything with increased stats, though (like the cafe/cospatio)
Also awp, in irc I said you'd need 21 battle+transport to get another stop, but it's actually 41, then you get another stop every 20 points