By my reading of the ability, it only grants bonus spells to sorcerers. Non-sorcerers specifically gain bloodline powers, not bonus spells, bonus feats (except through the Dragon Disciple's own bonus feat ability), or bloodline arcana.
However, that's a strict rules-as-written reading. The intent seems to be to allow you to gain those spells at the appropriate Dragon Disciple levels, and probably the rest of the bloodline stuff as well. Not getting the bloodline arcana just wouldn't make much sense. It's probably best to ask your GM to look at it, I think he or she is pretty likely to give you the bonus skill, spells, and bloodline arcana. (You would still only gain bonus feats through the prestige class's bonus feat ability.)
The main advantadge on any Martial character after level 5 is iterative attacks. Using a barbarian with these starts will definitely let you get an early start with reduced power as you level higher.
Shifter's Blessing doesn't kick in until Ranger level 3, which will give you 2 levels of Barbarian at level 5.
This means you'll have a total of 4 Feats + 1 rage power. I would suggest taking Power Attack at level 1, Rending Claws at 2nd, Extra Rage Power at 3rd (Assuming you have a level of Barbarian by now), and Amplified Rage at 5th.
Improved Natural Attack:Gore is worth looking into as your level 7 feat as well.
This will make sure that you do have Rend with your claws, and that while shifted and Raging, you will have a +13 to hit with all attacks, or +11 while Power Attacking.
This however won't let you gain extra attacks as your BAB increases at later levels, and there are a couple of issues.
Abyssal Blood is a level 6 power, and as such can only be taken by a character with 6 class levels in Barbarian.
The middle blood rage power (that without "lesser" or "greater" in the title) can only be taken by a 6th level or higher barbarian that already has the lesser blood rage power of its group.
If you were to compare your single attack damage against a fighter of a similar level, I think you would fall short of damage in a case by case basis, though as your build focuses on Full Attacks, this is an acceptable turnover.
On any given attack vs a monster with 17AC, you've got about a 25% chance of missing. Assuming you hit every attack, you'll be dealing around 70-80 damage with a full attack, and at most with one attack, you'll hit 26 with your Gore.
This AnyDice Link will show the average damage you can expect for each attack.
Best Answer
I think you are in house-rule territory. I'd suggest upgrading the die type on the claws for each feat which grants claws, so that the character had claws that do 1d8 damage each (1d4 for basic claws -> 1d6 -> 1d8).
Possibly, by RAW, the character would have just one set of 1d4 claws, although I cannot find a definitive answer. The simplest rationale is that claws don't stack in the same way that magic daggers don't stack (you don't get 5 attacks just because you have 5 +1 daggers). None of the rules suggest the character has grown extra limbs or the ability to make extra attacks.
The reason I suggest the house rule is that this character has invested quite a bit into multiple effects that grant claws, and it seems fair to give them something for it whilst the game rules say nothing.