Do I pay all the costs and choose all the targets when the spell is cast?
Yes.
Does the damage resolve sequentially or simultaneously?
All targets take the damage simultaneously.
Can I chain the lightning from target A to target B back to target A?
No, that's not what 'new' means. You can only choose new targets that haven't already been targeted by the spell.
Can additional targets be in different regions (as long as they're nearby the previous target)?
No. Being nearby the previous target is an additional targeting restriction, not a loosening of the targeting rules, so all targets must be in the same region as the caster.
