I am playing a Synthesist summoner level 7. I am thinking of taking 6 levels in skald to get the Linnorm Death Curse, Taiga rage power but i am not sure if it would activate while using raging song when my eidolon dies.
The skald rage power taiga linnorm death curse says, "The [affected] character’s melee attacks deal an additional 1 point of electricity damage. If the [affected] character is knocked unconscious or killed by an attack or spell, the attacker suffers the curse of electricity." The rage power then goes on to describe the curse of electricity: a failed Will saving throw means the attacker gains electricity vulnerability.
While a synthesist summoner's fused eidolon cannot be knocked unconscious, it can be killed: "The synthesist gains the eidolon’s hit points as temporary hit points [and w]hen these hit points reach 0, the eidolon is killed and sent back to its home plane." So if a skald's supernatural ability raging song is used for the supernatural ability inspired rage to grant the fused eidolon the rage power taiga linnorm death curse and the synthesist summoner's eidolon is subsequently killed, the rage power should function normally.
An Aside: This is extremely suboptimal
Unless either the party is similarly unoptimized or the summoner is vastly overshadowing the remainder of the party or everyone else in the party is throwing around huge amounts of electricity damage, there's very little reason to make this sort of investment. The rage power taiga linnorm death curse base effect (the extra electricity damage) is trivial (especially at the proposed level 13!), and the curse itself only has the possibility of affecting one enemy if an ally is removed from the encounter!
A typical summoner will be much better served by taking six more levels of summoner than by taking six levels of skald solely to acquire and share this rage power with his eidolon.
Yes, if your Eidolon is killed while under the effect of a Linnorm Death Curse rage power, the curse will be applied to whoever killed the Eidolon. The specific interaction of wording:
If the character is knocked unconscious or killed by an attack or spell, the attacker suffers the curse of electricity (Will negates).
The synthesist gains the eidolon’s hit points as temporary hit points. When these hit points reach 0, the eidolon is killed and sent back to its home plane.
A better question might be - why would you ever allow your fused eidolon to be killed in this way, when you can use Fused Link to sacrifice from your own hit points and keep it alive? Presumably being fused with your Eidolon is the basis of much of your combat effectiveness and it seems unlikely that giving an enemy a chance of becoming vulnerable to electricity is worth sacrificing the benefits that the fusion gives you. Fused Link allows you to keep the Eidolon alive until you must both go down.
No, it wouldn't
Your eidolon cannot be targeted by any effects while you two are a single creature and anything that would affect the eidolon will affect your character instead (with exceptions, see this answer).
Neither the synthesist nor his eidolon can be targeted separately, as they are fused into one creature.
This is explained on a FAQ item aswell.
So, if the eidolon cannot the targeted by your Inspired Rage, whenever it reaches 0 hit points and dies, there is no eletricity damage on the attacker because the eidolon wasn't affected by the rage power.