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So today in a game I used Orks as my Primary, and CSM as a second detachment Allies of Convenience.

Cards like More Dakka that calls for an enemy to be completely destroyed in the Shooting Phase, I could not see any restriction on it from only the Ork Faction...

Stomp em Boyz only calls for an enemy unit to be destroyed in your Assault phase, while More Speed, Go Fasta! specifically calls for Ork units...

So unless specifically stated, can Allies accomplish Specific Faction objective cards?

Depends on the card, but generally there's no rule against it.

Some of the faction decks will call out that they can only be achieved by units of a certain Faction (at least one of the Dark Angels ones is like this), whilst others don't care.

They are all, after all, part of your army.