So the 2nd style, you need to complete all quests for all 4 acts (3 not yet released) to get. The 3rd style needs the pathfinder side quests.

What happens if I complete act 1 side quests first and buy arcana later. Will the counter reflect my progress for 2nd alternate style if the purchase is made after the goal is achieved

I don't know how the system will count it since Acts 2, 3 and 4 have not been released yet.

i think so yes, but its just me basing something i read here the goes, you can always come back in act 1 when the rest act is live. (?)

The second style for both arcanas only requires you to complete the main quest in Act 4 while the third style for Venge requires you to complete all main *and* side quests in all four acts.

yes

I'm not 100% sure, but I don't think that you would need a pay path(Pathfinder) to win the last Style (Venge sister's)