First, your questions:
1) Considering you want to play a
kobold sorcerer going in to Rage Mage, I'd say we can optimize quite a bit... but your DM probably won't like it, even if the overall power level is around Tier 3.
2) At low levels, you'll be a superstar. Kobold + all the cheese necessary to make you a decent Rage Mage = fan-fricking-tastic. At mid-high levels, your power level will drop off significantly, but you'll still be able to pull your weight in the party.
3) Considering you're tied to Rage Mage, that's probably the best we're going to do. Maybe a level of Champion of Gwynharwyf or some other Rage-granting class instead of Barbarian, but what's the point?
4) Depends. For anyone other than a Kobold Sorcerer, upping it to 7 or 9/10 casting would make it better. But Kobolds get enough "+1 sorcerer caster level" freebies that it hardly matters. But Rage Mage is widely considered a "Down 2 PrC", so you'd be going from a Tier 1/2 to a Tier 3/4, so you might not need any changes.
5) None. Battle Sorc is good (though maybe not
as good in all situations), and Barbarian is probably the best 1 or 2 level dip for any melee class.
Now, the Build:
White Dragonspawn Kobold Battle Sorcerer 4/Barbarian 1/Rage Mage 10/Abjurant Champion 5
Feats: Dragonwrought, Combat Casting, Draconic Reservoir, Practiced Caster, whatever else. Extra Rage, I don't care.
The Trick:
Dragonspawn is a template from Dragonlance Campaign Setting (pg 222). Specifically, the White version only has +1 LA, and increases your effective Sorcerer level by 1. Note that Dragonspawn is a template that can only be applied to Humanoids or Monsterous Humanoids (which a Dragonwrought Kobold is not), but that's fine because you choose your race before you choose feats according to pg 11 of the PHB. When you later
Buy Off your level adjustment, you effectively gain +1 Sorc level for free. Undergo the
Greater Draconic Rite of Passage. That's another +1 effective Sorc level. Since you took Dragonwrought, you can become a Loredrake (Dragons of Eberron, pg 31) for another +2 Sorc levels. When Soro_Lost shows up to tell you this trick doesn't work, tell him to STFU. Nothing personal, Soro.
So with all of that, you net +4 effective Sorcerer levels. Between 10 levels of Rage Mage and a level of Barbarian, you lost 6. So you almost make up the difference, but still get 9th level spells (at level 20).
Now, with all that being said, I strongly advise disregarding everything I just said and
not taking 10 levels of Rage Mage. Seriously, the class's iconic ability comes at level 1. You could make a more traditional Gish like Battle Sorc 4/Barbarian 1/Rage Mage 1/Abjurant Champion 5/ Spellsword 1/Eldritch Knight the rest. You'd still need the Greater Draconic Rite to get 9th level spells, but that is
way less cheesy than relying on Dragonwrought -> Loredrake
and White Dragonspawn in the same build, especially since it requires your DM believing that Dragonwrought Kobolds are True Dragons (or that Lesser Dragons can take a Sovereign Archetype)
and mixing Eberron and Dragonlance material (some DMs don't like mixing settings).
