I have searched high and low for the answer to this but haven't been able to find anything definitive.
That is because the answers are not all definitive
If you are wearing +3 handwraps of mighty blows with 3 different property runes ie flaming, shocking, corrosive, would that get applied to your attacks after you transform into a dragon?
It varies
I know for sure that the striking runes can't increase it beyond 3 damage die. However barbarians dragon transformation says that you use your own AC and attack modifier.
The Handwraps apply to all unarmed attacks, however, three things;
1. I am unsure if it still applies after you transform into a dragon considering you change shape, however, it does say to use your own attack modifier, and the spell itself says you have hands in this form (and can take manipulate actions).
So you would use your own AC and to hit. The general polymorph rules don't allow you to use item bonuses while in a battle form like this dragon spell the special statistics can be adjusted only by circumstance bonuses, status bonuses, and penalties I believe that you can apply them because it figures into your to hit number and AC first, then you substitute it into the dragon form. (Some others disagree but I see no reason to give any weight to that opinion here because it would make this form a total curse that no barbarian would want)
2. If it does apply, does it apply completely? ie, some attacks have base damage of 2d_something + xd_something(elemental) will it increase the damage up to 3d_something + xd_something(elemental)?
The polymorph rules say the constant abilities of your gear still function so the hand wraps still apply once you are in the dragon form. Handwraps do seem to apply to all your unarmed attacks by their description.
The item bonus to attack obviously doesn't add from here but it has already been factored in so no problem.
I don't see that the striking runes do anything. The damage dice are part of the special statistics of the spell and so can't be modified by the striking rune. The spell is doing a base 2d12 piercing plus 2d6 acid in terms of dice - so that is OK compared to 3d12 you might be doing with your weapon at this level.
However flaming, shocking, corrosive are all additional damage and Mark has said that additional damage still applies. So those work.
So does your extra damage from raging (the feat reminds you of this) and the additional damage from weapon specialization.
I think you are doing reasonable damage not dissimilar to your greataxe, at the same attack and defense. But you are getting reach and a breath weapon plus you can fly.
3. If it does not apply at all isn't transforming into another shape completely worthless during combat as you are in the end missing out on up to +3 to all attack rolls and an extra 3d6 worth of base damage from elemental property runes (not including the crit bonuses)
Yes precisely missing out on +3 to hit would drop your damage output by 30-60%. No barbarian would bother with this apart from some minor utility - and that is just not what barbarians do.
Obviously this is my opinion, it leaves you with a reasonable ability. Ask someone else you will get a different answer. The rules are not tightly defined enough for us to lock in an answer.
1) I'd say yes, the +3 from handwraps applies. Because otherwise the sentence that you use your own attack bonus in Dragon Transformation makes no sense because that's already allowed with polymorph spells. So Dragon Transformation has a special rule there.
2) I'd say yes because elemental runes are additional/extra damage and without allowing that other features like Sneak Attack and Precision Edge would also not work while polymorphed.
The rules definitily need clarification on what is allowed while polymorphed. But as Gortle said not allowing the + from handwraps kills the transformation for barbarians.
I think the to hit bonus applies, but not the damage and maybe not the elemental runes either.
The only part I'm super confidant on is the to-hit bonus, as you substitute your attack bonus (which includes the item bonus) instead of the spell value if it's higher (as a special function of being a dragon barbarian).
what about if it had the major striking rune on?
Would it increase the dice count from 3 to 4 dice? I would assume greater striking would not change the bite dice amounts because it has the elemental damage added to it that is technically your 3rd die anyway.
The definition of striking says "increasing the weapon damage dice it deals to two instead of one", and obviously major striking increases it to 4 dice, I would assume that the bite itself is the weapon and the elemental damage is in this instance going to be included as part of that weapon. Hence it starts at 3* dice and can maybe be upgraded via the major striking rune??
The polymorph rules state
If you take on a battle form with a polymorph spell, the special statistics can be adjusted only by circumstance bonuses, status bonuses, and penalties
If you look at one of the battle form spells Dinosaur From they tell you what the special statistics are. These include the damage dice of the attack.
Which means you can't gain extra weapon dice on these attacks. So striking runes just don't work here, at all, ever. The other things are not part of the statistics. They are extra things not mentioned - so they do add.
However the language is totally vague. "Special Statistics" are nowhere precisely defined. So if you want to bend the wording a bit and can get it past your GM and you are both happy with that - then don't let me spoil your fun.
I wouldn't allow any Striking runes to modify battle form damages. It doesn't seem to be intended and it can really mess up damage numbers!
If you look up Dinosaur Form, with striking runes applied, some variants have 1 and some 2 damage dice. If you allow Striking runes, will only the 1 damage die variants get upgraded? So, no to Striking runes there, the Form spells don't work with those.