Re: Hypothetical questions
Not fighting at Karbala would have made no difference. The schism between Sunnis and Shias was well underway by that point because the Muslims were already divided between those who thought Khilfat should go to whoever the Ummah wanted, and those who thought it belonged only amongst the Ahle Bait.
Same deal with the war between Hazrat Ali (ra) and Bibi Ayesha (ra). It had no bearing on the Sunni/Shia split and resulting schism because factions amongst the Muslims had emerged long after that.