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Firstly, Power weapons. Units like Death Cult Assasins are capable of getting multiple power weapons for no cost (DCA get two power weapons, are WS4, S4, I5 and A2 base AFAIR) They have 2 power weapons base.

Aside from the fact that people like to call modeling a power weapon in a specific way to resemble one of the descriptions in the rulebook MFA, is it legal for the model in question to gain the extra attack for having 2 CCW as usual, but then benefit from the rules of both?

so a DCA with a power axe and a mace is I1 but S7, AP2. (S4+1 for axe +1 for Mace) or simply a maul and sword (I5 with S6AP3.)

Secondly, the wording of the "Specialist Weapon" rule is very open to interpretation.

It reads:

BRB, Page 43 wrote:A model with this special rule does not recieve +1 attacks for fighting with two weapons unless both weapons have the specialist special rule

So does this mean that one may use a Lighting claw and a power fist, and gain +1 to attacks? furthermore, if the profiles are combined, does the strike at I1 S*2 and have the shred special rule?

Thanks in advance.

About your first part of "getting the benefits of both weapons" you should refer to the page 51, where it tells you to choose one weapon if you have multiple cc weapons and use its profile.
You do get the bonus attack for having 2 CCW though.
Also, as you have noticed "specialist" weapons no longer require to use 2 of one kind of weapon like LC/PF/TH used to.
There too you choose with which weapon you fight, so either your claw or your fist, while getting an extra attack either way.

As above, you get to utilize the +1 attack for having two CCWs in both cases, but you have to pick one of the two weapon profiles your attacks will use in that particular round of combat. Next turn you can use the other profile if you wish.