You're asking a question(s), that by the very nature of the Alpha Legion, we simply won't be able to answer with any degree of certainty. Unless they decide to tell us more about their decision in the Horus Heresy series about the Primarchs decision(s).
The Cabal certainly seem to imply at one stage that Alpharius and Omegon are one soul within two bodies and they do seem to complete each others sentences, but then so do the Legion Captains, so this isn't conclusive either. Even if they are one soul, it doesn't seem as if they operate as one.
The Serpent Beneath seems to indicate that they are working against each other on some level, quite how far or to what extent isn't really shown. It might have been an isolated incident or it could be the the sign of a schism starting to form between them. Whether this is simply conflicting views on how the war should be fought, or as some fan theories are suggesting, one of the two is staying more loyal to the Emperor and working against the Heresy, where as the other one is still trying to ensure the Heresy succeeds as per the Cabal.
From both Sheed Rankos closing words in Legion, 'For the Emperor', in answer as to why they are attacking the fleet, and Alpharius asserting to the Cabal that his loyalty is to the Emperor always after they are shown the Acuity. It certainly seems that they are following the Cabals vision. But then their actions in Deliverance Lost at the end leave that in doubt.
As for the Legions current actions in 40k, no answer could be given at all really. It's just impossible to know what the hell they are doing. One or both of the Primarchs could easily be alive or both dead. They or another Alpha Legionnaire could be leading the entire Legion with to support chaos, their own agenda, someone else or even the Imperium for all we know. Or like you have suggested, the Legion is entirely fractured and just operating how ever the fuck they want.
Only one thing is certain.
I am Alpharius.